558 results on '"Graph"'
Search Results
2. A Collaborative Graph-based SLAM Framework Using a Computationally Effective Measurement Algebra.
- Author
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Péter, Gábor and Kiss, Bálint
- Subjects
ALGEBRA ,SENSOR placement ,COMPUTER performance ,DATABASES - Abstract
Simultaneous localization and mapping (SLAM) is an essential task for autonomous rover navigation in an unknown environment, especially if no absolute location information is available. This paper presents a computationally lightweight framework to enable agents with limited processing power to carry out the SLAM cooperatively and without absolute onboard localization sensors in a 2D environment. The proposed solution is built on a graph-based map representation, where nodes (resp. edges) represent landmarks (resp. odometry-based relative measurements), a measurement algebra with embedded uncertainty, and a compact database format that could be stored on a server in a centralized manner. The operations required by the agents to insert a new landmark in the graph, update landmark positions and combine measurements as a loop is closed in the graph are detailed. The resulting framework was tested in a laboratory environment and on a public dataset with encouraging results; hence our method can be used for costeffective indoor mobile agents with limited computational resources and onboard sensors to achieve a mapping while keeping track of the agent's position. The method can also be easily generalized for a 3D scenario. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. Function Analysis
- Author
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Sobot, Robert and Sobot, Robert
- Published
- 2023
- Full Text
- View/download PDF
4. Leibniz algebras and graphs.
- Author
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Barreiro, Elisabete, Calderón, Antonio J., Lopes, Samuel A., and Sánchez, José M.
- Subjects
- *
LIE algebras , *ALGEBRA , *SUBGRAPHS , *STRUCTURAL analysis (Engineering) , *GROBNER bases - Abstract
We consider a Leibniz algebra L = I ⊕ V over an arbitrary base field F , being I the ideal generated by the products [ x , x ] , x ∈ L . This ideal has a fundamental role in the study presented in our paper. A basis B = { v i } i ∈ I of L is called multiplicative if for any i , j ∈ I we have that [ v i , v j ] ∈ F v k for some k ∈ I. We associate an adequate graph Γ (L , B) to L relative to B . By arguing on this graph we show that L decomposes as a direct sum of ideals, each one being associated to one connected component of Γ (L , B). Also the minimality of L and the division property of L are characterized in terms of the weak symmetry of the defined subgraphs Γ (L , B I ) and Γ (L , B V ). [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
5. Graphs Based on Equality Algebras.
- Author
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Kologani, M. Aaly, Muhiuddin, G., Borzooei, R. A., and Rezaei, G. R.
- Subjects
- *
ALGEBRA , *RELATION algebras , *PLANAR graphs , *COMPLETE graphs , *BIPARTITE graphs , *DIVISOR theory - Abstract
In this paper, we introduce new kinds of graphs based on equality algebras. First of all, by using the meet operation we define the notion of zero divisors on equality algebra and study related properties. Then we introduce a meet graph on equality algebra by using zero divisors. In addition, we investigate some conditions for a meet graph to be a complete, connected and a star graph. Also, by using the notion of filters in equality algebras, we define an equivalence relation on equality algebras and then by using the equivalence classes we introduce two kinds of graphs on them. Finally, conditions for a graph based on these classes to be connected, or bipartite, or complete, or planar or an outer-planar graph are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
6. Fundamentals of fractional revival in graphs.
- Author
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Chan, Ada, Coutinho, Gabriel, Drazen, Whitney, Eisenberg, Or, Godsil, Chris, Kempton, Mark, Lippner, Gabor, Tamon, Christino, and Zhan, Hanmeng
- Subjects
- *
OPEN-ended questions , *ALGEBRA , *GENERALIZATION - Abstract
We develop a general spectral framework to analyze quantum fractional revival in quantum spin networks. In particular, we determine when the adjacency algebra of a graph contains a matrix of a block diagonal form required for fractional revival, and introduce generalizations of the notions of cospectral and strongly cospectral vertices to arbitrary subsets of vertices. We give several constructions of graphs admitting fractional revival. This work resolves two open questions of Chan et al. (2019) [6]. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
7. Derivations of evolution algebras associated to graphs over a field of any characteristic.
- Author
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Reis, Tiago and Cadavid, Paula
- Subjects
- *
ALGEBRA , *FINITE, The , *LIE algebras - Abstract
The space of derivations of finite dimensional evolution algebras associated to graphs over a field with characteristic zero has been completely characterized in the literature. In this work we generalize that characterization by describing the derivations of this class of algebras for fields of any characteristic. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
8. Finite-dimensional Zinbiel algebras and combinatorial structures.
- Author
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Ceballos, Manuel, Núñez, Juan, and Tenorio, Ángel F.
- Subjects
- *
ALGEBRA , *ISOMORPHISM (Mathematics) , *ALGORITHMS - Abstract
In this paper, we study the link between finite-dimensional Zinbiel algebras and combinatorial structures or (pseudo)digraphs determining which configurations are associated with those algebras. Some properties of Zinbiel algebras that can be read from their associated combinatorial structures are studied. We also analyze the isomorphism classes for each configuration associated with these algebras providing a new method to classify them and we compare our results with the current classifications of 2- and 3-dimensional Zinbiel algebras. We also obtain the 3-vertices combinatorial structures associated with such algebras. In order to complement the theoretical study, we have designed and performed the implementation of an algorithm which constructs and draws the (pseudo)digraph associated with a given Zinbiel algebra and, conversely, another procedure to test if a given combinatorial structure is associated with some Zinbiel algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
9. Sensorimotor Underpinnings of Mathematical Imagination: Qualitative Analysis.
- Author
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McCollum, Gin
- Subjects
MENTAL imagery ,VESTIBULAR apparatus ,IMAGINATION ,POSTURE ,SPACE environment - Abstract
Many mathematicians have a rich internal world of mental imagery. Using elementary mathematical skills, this study probes the mathematical imagination's sensorimotor foundations. Mental imagery is perturbed using body position: having the head and vestibular system in different positions with respect to gravity. No two mathematicians described the same imagery. Eight out of 11 habitually visualize, one uses sensorimotor imagery, and two do not habitually used mental imagery. Imagery was both intentional and partly autonomous. For example, coordinate planes rotated, drifted, wobbled, or slid down from vertical to horizontal. Parabolae slid into place or, on one side, a parabola arm reached upward in gravity. The sensorimotor foundation of imagery was evidenced in several ways. The imagery was placed with respect to the body. Further, the imagery had a variety of relationships to the body, such as the body being the coordinate system or the coordinate system being placed in front of the eyes for easy viewing by the mind's eye. The mind's eye, mind's arm, and awareness almost always obeyed the geometry of the real eye and arm. The imagery and body behaved as a dyad, so that the imagery moved or placed itself for the convenience of the mind's eye or arm, which in turn moved to follow the imagery. With eyes closed, participants created a peripersonal imagery space, along with the peripersonal space of the unseen environment. Although mathematics is fundamentally abstract, imagery was sometimes concrete or used a concrete substrate or was placed to avoid being inside concrete objects, such as furniture. Mathematicians varied in the numbers of components of mental imagery and the ways they interacted. The autonomy of the imagery was sometimes of mathematical interest, suggesting that the interaction of imagery habits and autonomy can be a source of mathematical creativity. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
10. Sensorimotor Underpinnings of Mathematical Imagination: Qualitative Analysis
- Author
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Gin McCollum
- Subjects
embodied cognition ,mental Imagery ,mathematics education ,graph ,algebra ,sensorimotor ,Psychology ,BF1-990 - Abstract
Many mathematicians have a rich internal world of mental imagery. Using elementary mathematical skills, this study probes the mathematical imagination's sensorimotor foundations. Mental imagery is perturbed using body position: having the head and vestibular system in different positions with respect to gravity. No two mathematicians described the same imagery. Eight out of 11 habitually visualize, one uses sensorimotor imagery, and two do not habitually used mental imagery. Imagery was both intentional and partly autonomous. For example, coordinate planes rotated, drifted, wobbled, or slid down from vertical to horizontal. Parabolae slid into place or, on one side, a parabola arm reached upward in gravity. The sensorimotor foundation of imagery was evidenced in several ways. The imagery was placed with respect to the body. Further, the imagery had a variety of relationships to the body, such as the body being the coordinate system or the coordinate system being placed in front of the eyes for easy viewing by the mind's eye. The mind's eye, mind's arm, and awareness almost always obeyed the geometry of the real eye and arm. The imagery and body behaved as a dyad, so that the imagery moved or placed itself for the convenience of the mind's eye or arm, which in turn moved to follow the imagery. With eyes closed, participants created a peripersonal imagery space, along with the peripersonal space of the unseen environment. Although mathematics is fundamentally abstract, imagery was sometimes concrete or used a concrete substrate or was placed to avoid being inside concrete objects, such as furniture. Mathematicians varied in the numbers of components of mental imagery and the ways they interacted. The autonomy of the imagery was sometimes of mathematical interest, suggesting that the interaction of imagery habits and autonomy can be a source of mathematical creativity.
- Published
- 2022
- Full Text
- View/download PDF
11. On the isomorphisms between evolution algebras of graphs and random walks.
- Author
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Cadavid, Paula, Rodiño Montoya, Mary Luz, and Rodriguez, Pablo M.
- Subjects
- *
RANDOM graphs , *NONASSOCIATIVE algebras , *ALGEBRA , *GROUP algebras , *ISOMORPHISM (Mathematics) , *AUTOMORPHISM groups , *RANDOM walks - Abstract
Evolution algebras are non-associative algebras inspired from biological phenomena, with applications to or connections with different mathematical fields. There are two natural ways to define an evolution algebra associated to a given graph. While one takes into account only the adjacencies of the graph, the other includes probabilities related to the symmetric random walk on the same graph. In this work we state new properties related to the relation between these algebras, which is one of the open problems in the interplay between evolution algebras and graphs. On the one hand, we show that for any graph both algebras are strongly isotopic. On the other hand, we provide conditions under which these algebras are or are not isomorphic. For the case of finite non-singular graphs we provide a complete description of the problem, while for the case of finite singular graphs we state a conjecture supported by examples and partial results. The case of graphs with an infinite number of vertices is also discussed. As a sideline of our work, we revisit a result existing in the literature about the identification of the automorphism group of an evolution algebra, and we give an improved version of it. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
12. Scaffolds: A graph-theoretic tool for tensor computations related to Bose-Mesner algebras.
- Author
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Martin, William J.
- Subjects
- *
ALGEBRA , *VECTOR spaces , *MATRIX multiplications , *PLANAR graphs , *COMBINATORICS - Abstract
We introduce a pictorial notation for certain tensors arising in the study of association schemes, based on earlier ideas of Terwilliger, Neumaier and Jaeger. These tensors, which we call "scaffolds", obey a simple set of rules which generalize common linear-algebraic operations such as trace, matrix product and entrywise product. We first study an elementary set of "moves" on scaffolds and illustrate their use in combinatorics. Next we re-visit results of Dickie, Suzuki and Terwilliger. The main new results deal with the relationships among vector spaces of scaffolds with edge weights chosen from a fixed coherent algebra and various underlying diagrams. As one consequence, we provide simple descriptions of the Terwilliger algebras of triply regular and dually triply regular association schemes. We finish with a conjecture connecting the duality of Bose-Mesner algebras to the graph-theoretic duality of circular planar graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
13. Evolution algebras, automorphisms, and graphs.
- Author
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Elduque, Alberto and Labra, Alicia
- Subjects
- *
ALGEBRA , *LIE algebras , *DIRECTED graphs , *AUTOMORPHISM groups - Abstract
The affine group scheme of automorphisms of an evolution algebra E with E 2 = E is shown to lie in an exact sequence 1 → D → A u t (E) → S , where D , diagonalizable, and S , constant, depend solely on the directed graph associated to E . As a consequence, the Lie algebra of derivations D e r (E) (with E 2 = E ) is shown to be trivial if the characteristic of the ground field is 0 or 2, and to be abelian, with a precise description, otherwise. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
14. Measuring lexicographic product network efficiency with small time delay.
- Author
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Li, Feng, Maseleno, Andino, Yuan, Xiaohui, and Balas, Valentina E.
- Subjects
- *
GRAPH theory , *TELECOMMUNICATION systems , *INFORMATION measurement , *ALGEBRA - Abstract
The diameter and distance parameters of a network play very significant roles in analyzing the efficiency of a communication network, these parameters provide some efficient ways to measure information time delay in communication networks. We use the lexicographic product method to construct a larger network model, which is called the lexicographic product network by some specified small graphs. Network models based on the lexicographic product method contain these small graphs as sub-networks, and many desirable properties of these sub-networks are preserved. By using algebra graph theory, we investigated the diameter parameters of the lexicographic product network, and established an enumeration formula which only depends on the parameters of sub-networks. By analyzing the diameter formula and comparing it with other network models, it is proved that the lexicographic product network has a smaller time delay. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
15. Characterization theorems for the spaces of derivations of evolution algebras associated to graphs.
- Author
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Cadavid, Paula, Rodiño Montoya, Mary Luz, and Rodriguez, Pablo M.
- Subjects
- *
ALGEBRA , *MATRICES (Mathematics) , *SPACE - Abstract
It is well-known that the space of derivations of n-dimensional evolution algebras with non-singular matrices is zero. On the other hand, the space of derivations of evolution algebras with matrices of rank n−1 have also been completely described in the literature. In this work, we provide a complete description of the space of derivations of evolution algebras associated to graphs, depending on the twin partition of the graph. For graphs without twin classes with at least three elements, we prove that the space of derivations of the associated evolution algebra is zero. Moreover, we describe the spaces of derivations for evolution algebras associated to the remaining families of finite graphs. It is worth pointing out that our analysis includes examples of finite dimensional evolution algebras with matrices of any rank. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
16. NETWORK MODELS.
- Author
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BAEZ, JOHN C., FOLEY, JOHN, MOELLER, JOE, and POLLARD, BLAKE S.
- Subjects
- *
ALGEBRA , *PROCEEDS , *CONSTRUCTION - Abstract
Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce 'network models' to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to Cat, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction. [ABSTRACT FROM AUTHOR]
- Published
- 2020
17. The connection between evolution algebras, random walks and graphs.
- Author
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Cadavid, Paula, Rodiño Montoya, Mary Luz, and Rodriguez, Pablo M.
- Subjects
- *
RANDOM graphs , *ABSTRACT algebra , *ALGEBRA , *NONASSOCIATIVE algebras , *PHENOMENOLOGICAL biology - Abstract
Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
18. Automorphism groups of Cayley evolution algebras
- Author
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Costoya, Cristina, Muñoz, Vicente, Tocino, Alicia, and Viruel, Antonio
- Subjects
Algebra and Number Theory ,Evolution algebra ,Applied Mathematics ,Mathematics - Rings and Algebras ,Graph ,Automorphism group ,Computational Mathematics ,Mathematics::Group Theory ,05C25, 17A36, 17D99 ,Rings and Algebras (math.RA) ,Álgebra ,FOS: Mathematics ,Geometry and Topology ,Finite group ,Analysis - Abstract
In this paper we introduce a new species of evolution algebras that we call Cayley evolution algebras. We show that if a field $k$ contains sufficiently many elements (for example if $k$ is infinite) then every finite group $G$ is isomorphic to $Aut(X)$ where $X$ is a finite-dimensional absolutely simple Cayley evolution $k$-algebra., Comment: This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is published under the Creative Commons Attribution license and can be freely dowloaded it from https://doi.org/10.1007/s13398-023-01414-w
- Published
- 2023
19. Techniques of graphs in the study of the structure of graded modules.
- Author
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Calderón Martín, Antonio J. and Navarro Izquierdo, Francisco J.
- Subjects
- *
NONASSOCIATIVE algebras , *MODULES (Algebra) , *VECTOR spaces , *INDECOMPOSABLE modules , *STRUCTURAL analysis (Engineering) - Abstract
We associate an adequate graph to any pair (V , W) where V is a graded module over a graded linear space W , in such a way that allows us to study the inner algebraic structure of (V , W). In particular, the homogeneous indecomposability, the homogeneous semisimplicity and the homogeneous simplicity of (V , W) are characterized in terms of this graph. Some applications to the theory of non-associative algebras and modules over algebras are also provided. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
20. Bilinear maps and graphs.
- Author
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Martín, Antonio J. Calderón and Izquierdo, Francisco J. Navarro
- Subjects
- *
VECTOR spaces , *INVARIANT subspaces - Abstract
Let V be a linear space of arbitrary dimension and over an arbitrary base field F , endowed with a bilinear map f : V × V → V. A basis B = { v i } i ∈ I of V is an f -basis if for any i , j ∈ I we have that f (v i , v j) ∈ F v k for some k ∈ I. We associate to any triplet (V , f , B) an adequate graph (V , E). By arguing on this graph we show that V decomposes as a direct sum of strongly f -invariant linear subspaces, each one being associated to one connected component of (V , E). Also the B -semisimplicity and the B -simplicity of V are characterized in terms of the associated graph. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
21. A polynomial recognition of unit forms using graph-based strategies.
- Author
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Alves, Jesmmer, Castongay, Diane, and Brüstle, Thomas
- Subjects
- *
POLYNOMIALS , *GRAPHIC methods , *ALGORITHMS , *GEOMETRIC vertices , *ALGEBRA - Abstract
Abstract The units forms are algebraic expressions that have important role in representation theory of algebras. We identified that existing algorithms have exponential time complexity for weakly nonnegative and weakly positive types. In this paper we introduce a polynomial algorithm for the recognition of weakly nonnegative unit forms. The related algorithm identifies hypercritical restrictions in a given unit form, testing every subgraph of 9 vertices of the unit form associated graph. By adding Depth First Search approach, a similar strategy could be used in the recognition of weakly positive unit forms. We also present the most popular methods to decide whether or not a unit form is weakly nonnegative or weakly positive, we analyze their time complexity and we compare the results with our algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
22. Single pushout rewriting in comprehensive systems of graph-like structures
- Author
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Patrick Stünkel and Harald König
- Subjects
Graph rewriting ,Property (philosophy) ,General Computer Science ,Computer science ,Pushout ,upper adjoint ,comprehensive system ,VDP::Matematikk og Naturvitenskap: 400::Informasjons- og kommunikasjonsvitenskap: 420 ,hereditary pushout ,Graph ,Theoretical Computer Science ,Algebra ,partial morphism ,category theory ,Range (mathematics) ,Morphism ,single pushout rewriting ,Homogeneous ,Mathematics::Category Theory ,Rewriting - Abstract
The elegance of the single-pushout (SPO) approach to graph transformations arises from substituting total morphisms by partial ones in the underlying category. SPO's applicability depends on the durability of pushouts after this transition. There is a wide range of work on the question when pushouts exist in categories with partial morphisms starting with the pioneering work of Lowe and Kennaway and ending with an essential characterisation in terms of an exactness property (for the interplay between pullbacks and pushouts) and an adjointness condition (w.r.t. inverse image functions) by Hayman and Heindel. Triple graphs and graph diagrams are frameworks to synchronise two or more updatable data sources by means of internal mappings, which identify common sub-structures. Comprehensive systems generalise these frameworks, treating the network of data sources and their structural inter-relations as a homogeneous comprehensive artefact, in which partial maps identify commonalities. Although this inherent partiality produces amplified complexity, we can show that Heindel's characterisation still yields existence of pushouts in the category of comprehensive systems and reflective partial morphisms and thus enables computing by typed SPO graph transformation.
- Published
- 2021
23. Cohen-Macaulay Properties of Closed Neighborhood Ideals
- Author
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Leaman, Jackson
- Subjects
- Closed neighborhood ideal, Cohen Macaulay, Squarefree monomial ideal, Graph, Complete intersection ideal, Algebra, Discrete Mathematics and Combinatorics
- Abstract
This thesis investigates Cohen-Macaulay properties of squarefree monomial ideals, which is an important line of inquiry in the field of combinatorial commutative algebra. A famous example of this is Villareal’s edge ideal [11]: given a finite simple graph G with vertices x1, . . . , xn, the edge ideal of G is generated by all the monomials of the form xixj where xi and xj are adjacent in G. Villareal’s characterization of Cohen-Macaulay edge ideals associated to trees is an often-cited result in the literature. This was extended to chordal and bipartite graphs by Herzog, Hibi, and Zheng in [7] and by Herzog and Hibi in [6]. In 2020, Sharifan and Moradi [10] introduced a related construction called the closed neighborhood ideal of a graph. Whereas an edge ideal of a graph G is generated by monomials associated to each edge in G, the closed neighborhood ideal is generated by monomials associated to its closed neighborhoods. In 2021, Sather-Wagstaff and Honeycutt [8] characterized trees whose closed neighborhood ideals are Cohen-Macaulay. We will provide a generalization of this characterization to chordal graphs and bipartite graphs. Additionally, we will survey the behavior of the depth of closed neighborhood ideals under certain graph operations.
- Published
- 2023
24. Some operations on Dombi neutrosophic graph
- Author
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Arindam Dey, Kartick Mohanta, Tejinder Singh Lakhwani, Anita Pal, and Sankar Prasad Mondal
- Subjects
0209 industrial biotechnology ,General Computer Science ,Mathematics::General Mathematics ,Computer science ,Intersection (set theory) ,Computational intelligence ,Vagueness ,02 engineering and technology ,Extension (predicate logic) ,Cartesian product ,Graph ,Algebra ,symbols.namesake ,020901 industrial engineering & automation ,Product (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,MathematicsofComputing_DISCRETEMATHEMATICS ,Complement (set theory) - Abstract
Neutrosophic graph is an extension of fuzzy graph and intuitionistic fuzzy graph. It is helpful to handle the uncertainty (vagueness), related with the information of any decision making problem, where intuitionistic fuzzy graph models may fail to model properly. The Dombi operators have some excellent efficiency and flexibility to work with complex decision making problems. Utilizing these two ideas, we introduce a novel concept of Dombi neutrosophic graph in this manuscript. Some different types of Dombi neutrosophic graph such as a regular Dombi neutrosophic graph, strong Dombi neutrosophic graph, complete Dombi neutrosophic graph, and complement Dombi neutrosophic graph are presented and some properties are also described. We also define some different operations on Dombi neutrosophic graphs; viz. union, intersection composition, cartesian product, boxdot product, homomorphic product and modular product. An application of Dombi neutrosophic graph is also described in this manuscript.
- Published
- 2021
25. A note on different types of product of neutrosophic graphs
- Author
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Arindam Dey, Anita Pal, and Kartick Mohanta
- Subjects
Total degree ,020209 energy ,Fuzzy set ,Neutrosophic set ,Computational intelligence ,Vagueness ,02 engineering and technology ,General Medicine ,Graph ,Algebra ,Product (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Symmetric difference ,Mathematics - Abstract
Fuzzy set and neutrosophic set are two efficient tools to handle the uncertainties and vagueness of any real-world problems. Neutrosophic set is more capable than fuzzy set to deal the uncertainties of a real-life problem. This research paper introduces some new concept of single-valued neutrosophic graph (SVNG). We have also presented some different operations on SVNG such as rejection, symmetric difference, maximal product, and residue product with appropriate examples, and some of their important theorems are also described. Then, we have described the concept of total degree of a neutrosophic graph with some interesting examples. We have also presented an efficient approach to solve a decision-making problem using SVNG.
- Published
- 2021
26. On the Radical of a Hecke–Kiselman Algebra
- Author
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Magdalena Wiertel and Jan Okniński
- Subjects
Mathematics::Number Theory ,General Mathematics ,Modulo ,010102 general mathematics ,Semiprime ,Mathematics - Rings and Algebras ,Rational function ,01 natural sciences ,Graph ,010101 applied mathematics ,Algebra ,Rings and Algebras (math.RA) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,0101 mathematics ,Direct product ,Mathematics - Abstract
The Hecke-Kiselman algebra of a finite oriented graph $\Theta$ over a field $K$ is studied. If $\Theta$ is an oriented cycle, it is shown that the algebra is semiprime and its central localization is a finite direct product of matrix algebras over the field of rational functions $K(x)$. More generally, the radical is described in the case of PI-algebras, and it is shown that it comes from an explicitly described congruence on the underlying Hecke-Kiselman monoid. Moreover, the algebra modulo the radical is again a Hecke-Kiselman algebra and it is a finite module over its center., Comment: 11 pages
- Published
- 2020
27. On Linear Functions and Their Graphs: Refining the Cartesian Connection
- Author
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Rina Zazkis and Leslie Glen
- Subjects
Computer science ,General Mathematics ,Physics::Physics Education ,Science education ,Graph ,Education ,law.invention ,Algebra ,Visual inspection ,law ,Algebra representation ,Mathematics education ,Cartesian coordinate system ,Algebraic number ,Remedial education ,Linear equation - Abstract
This study focuses on connections between linear functions and their graphs that were made by tertiary remedial algebra students. In particular, we describe students’ work on a Task designed to examine the connection between points on a graph and the equation of a line. The data consist of 63 responses to a written questionnaire and individual interviews with three participants. The results indicate that visual approaches impede students’ solutions and point to incomplete connections between algebraic and graphical representation. While algebraic approaches point to various connections used in approaching the Task, students’ ability to work with algebraic representation did not necessarily result in capitalizing on these connections. Furthermore, interpretations of the graph based on visual inspection appeared most useful when used in support of the algebraic approach.
- Published
- 2020
28. The zero-divisor graph of an amalgamated algebra
- Author
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M. R. Doustimehr and Y. Azimi
- Subjects
Algebra ,Mathematics::Commutative Algebra ,Ring homomorphism ,General Mathematics ,Commutative ring ,Subring ,Graph ,Zero divisor ,Mathematics - Abstract
Let R and S be commutative rings with identity, $$f:R\rightarrow S$$ a ring homomorphism and J an ideal of S. Then the subring $$R\bowtie ^fJ:=\{(r,f(r)+j)\mid r\in R$$ and $$j\in J\}$$ of $$R\times S$$ is called the amalgamation of R with S along J with respect to f. In this paper, we generalize and improve recent results on the computation of the diameter of the zero-divisor graph of amalgamated algebras and obtain new results. In particular, we provide new characterizations for completeness of the zero-divisor graph of amalgamated algebra, as well as, a complete description for the diameter of the zero-divisor graph of amalgamations in the special case of finite rings.
- Published
- 2020
29. On the Terwilliger algebra of distance-biregular graphs
- Author
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Štefko Miklavič and Blas Fernandez
- Subjects
Vertex (graph theory) ,Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,Valency ,010103 numerical & computational mathematics ,01 natural sciences ,Complete bipartite graph ,Graph ,Algebra ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,0101 mathematics ,Mathematics - Abstract
Let Γ denote a distance-biregular graph with vertex set X. Fix x ∈ X and let T = T ( x ) denote the Terwilliger algebra of Γ with respect to x. In this paper we consider irreducible T-modules with endpoint 1. We show that there are no such modules if and only if Γ is the complete bipartite graph K 1 , n ( n ≥ 1 ) and x is a vertex of Γ with valency 1. If the valency of x is at least 2 then we show that up to isomorphism there is a unique irreducible T-module of endpoint 1, and this module is thin.
- Published
- 2020
30. Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri
- Author
-
Fadime Ulusoy
- Subjects
Algebra ,Nonlinear system ,Variables ,media_common.quotation_subject ,education ,Function (mathematics) ,Graph ,Task (project management) ,Mathematics ,media_common - Abstract
This study aims to reveal how prospective teachers express the relationships between variables through graphical representations when interpreting dynamic functional situations involving two simultaneously changing quantities. 100 prospective middle school mathematics teachers participated to this case study. The data consisted of prospective teachers’ written responses to the task involving filling bottles with water and the graphs of volume as a function of height and clinical interviews were used to examine their covariational reasoning and graphing abilities. The findings showed that only six prospective teachers' graphical representations were correct for both dynamic functional situations. The most significant and common problems in the graphical representations were found such as (i) inability to coordinate slopes for linear relationships between variables, (ii) representing nonlinear relations of variables as linear relations, (iii) reversing the roles of dependent and independent variables, (iv) representing the relationship between variables as decreasing rather than increasing and (v) representing the relationships between variables to include more or less partitions to the graph than required by the given dynamic functional situation.
- Published
- 2020
31. The orthogonality relation classifies finite-dimensional formally real simple Jordan algebras
- Author
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Nik Stopar, Gregor Dolinar, and Bojan Kuzma
- Subjects
Algebra ,Algebra and Number Theory ,Jordan algebra ,Mathematics::Rings and Algebras ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Graph ,Clique number ,Mathematics - Abstract
It is shown that a finite-dimensional formally real simple Jordan algebra is completely determined by the relation of Jordan-orthogonality.Communicated by Prof. Alberto Elduque
- Published
- 2020
32. s-Step Enlarged Krylov Subspace Conjugate Gradient Methods
- Author
-
Sophie Moufawad
- Subjects
Physics::Computational Physics ,Iterative method ,Applied Mathematics ,Domain decomposition methods ,Krylov subspace ,Supercomputer ,Computer Science::Numerical Analysis ,Graph ,Mathematics::Numerical Analysis ,Algebra ,Computational Mathematics ,Conjugate gradient method ,Linear algebra ,Computer Science::Mathematical Software ,Mathematics - Abstract
Recently, enlarged Krylov subspace methods, which consist of enlarging the Krylov subspace by a maximum of $t$ vectors per iteration based on the domain decomposition of the graph of $A$, were intr...
- Published
- 2020
33. Adinkras: Graphs of Clifford Algebra Representations, Supersymmetry, and Codes
- Author
-
Kevin Iga
- Subjects
High Energy Physics - Theory ,Applied Mathematics ,Clifford algebra ,FOS: Physical sciences ,Context (language use) ,Mathematical Physics (math-ph) ,Algebraic geometry ,Supersymmetry ,Graph ,Algebra ,High Energy Physics - Theory (hep-th) ,15A67, 16G30, 05C25, 81T60 ,FOS: Mathematics ,Error correcting ,Algebraic topology (object) ,Representation Theory (math.RT) ,Mathematical Physics ,Mathematics - Representation Theory ,Mathematics - Abstract
An Adinkra is a graph from the study of supersymmetry in particle physics, but it can be adapted to study Clifford algebra representations. The graph in this context is called a Cliffordinkra, and puts some standard ideas in Clifford algebra representations in a geometric and visual context. In the past few years there have been developments in Adinkras that have shown how they are connected to error correcting codes, algebraic topology, algebraic geometry, and combinatorics. These connections also arise for Cliffordinkras. This paper introduces Cliffordinkras and describes the relationship to these subjects in that context. No previous knowledge of Adinkras and supersymmetry is assumed., Paper presented at the 12th International Conference on Clifford Algebras and their Applications in Mathematical Physics, August 2020, online conference, 24 pages, color figures/diagrams
- Published
- 2021
34. An algebraic theory of graph reduction
- Author
-
Arnborg, Stefan, Courcelle, Bruno, Proskurowski, Andrzej, Seese, Detlef, Goos, Gerhard, editor, Hartmanis, Juris, editor, Ehrig, Hartmut, editor, Kreowski, Hans-Jörg, editor, and Rozenberg, Grzegorz, editor
- Published
- 1991
- Full Text
- View/download PDF
35. Drags: A compositional algebraic framework for graph rewriting
- Author
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Jean-Pierre Jouannaud, Nachum Dershowitz, Tel Aviv University (TAU), Deduction modulo, interopérabilité et démonstration automatique (DEDUCTEAM), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Spécification et Vérification (LSV), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay), Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Université Paris-Saclay, Deducteam, and Tel Aviv University [Tel Aviv]
- Subjects
Vertex (graph theory) ,General Computer Science ,Computer science ,Generalization ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Term algebra ,ACM: G.: Mathematics of Computing ,Theoretical Computer Science ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Drags ,0202 electrical engineering, electronic engineering, information engineering ,[INFO]Computer Science [cs] ,Algebraic number ,Graph rewriting ,symmetric monoidal category ,Symmetric monoidal category ,Function (mathematics) ,Graph ,rewriting ,Vertex (geometry) ,Algebra ,010201 computation theory & mathematics ,Computer Science::Programming Languages ,020201 artificial intelligence & image processing ,Rewriting ,[INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC] ,Graphs ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
International audience; We are interested in a natural generalization of term-rewriting techniques to what we call drags, viz. finite, directed, ordered, rooted multigraphs, each vertex of which is labeled by a function symbol. To this end, we develop a rich algebra of drags that generalizes the familiar term algebra and its associated rewriting capabilities. Viewing graphs as terms provides an initial building block for rewriting with such graphs, one that should impact the many areas where computations take place on graphs.
- Published
- 2019
36. Applications of Graph Kannan Mappings to the Damped Spring-Mass System and Deformation of an Elastic Beam
- Author
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Mudasir Younis, Adrian Petruşel, and Deepak Singh
- Subjects
Elastic beam ,Article Subject ,Computer science ,lcsh:Mathematics ,010102 general mathematics ,Fixed-point theorem ,Graph theory ,Fixed point ,lcsh:QA1-939 ,01 natural sciences ,Graph ,010101 applied mathematics ,Algebra ,Nonlinear system ,Metric space ,Modeling and Simulation ,0101 mathematics - Abstract
The purpose of this article is twofold. Firstly, combining concepts of graph theory and of fixed point theory, we will present a fixed point result for Kannan type mappings, in the framework of recently introduced, graphicalb-metric spaces. Appropriate examples of graphs validate the established theory. Secondly, our focus is to apply the proposed results to some nonlinear problems which are meaningful in engineering and science. Some open problems are proposed.
- Published
- 2019
37. Strong forms of stability from flag algebra calculations
- Author
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Jakub Sliacan, Konstantinos Tyros, and Oleg Pikhurko
- Subjects
Computation ,010102 general mathematics ,0102 computer and information sciences ,Mathematical proof ,01 natural sciences ,Graph ,Theoretical Computer Science ,Combinatorics ,Algebra ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,QA ,Mathematics - Abstract
Given a hereditary family $\mathcal{G}$ of admissible graphs and a function $\lambda(G)$ that linearly depends on the statistics of order-$\kappa$ subgraphs in a graph $G$, we consider the extremal problem of determining $\lambda(n,\mathcal{G})$, the maximum of $\lambda(G)$ over all admissible graphs $G$ of order $n$. We call the problem perfectly $B$-stable for a graph $B$ if there is a constant $C$ such that every admissible graph $G$ of order $n\ge C$ can be made into a blow-up of $B$ by changing at most $C(\lambda(n,\mathcal{G})-\lambda(G)){n\choose2}$ adjacencies. As special cases, this property describes all almost extremal graphs of order $n$ within $o(n^2)$ edges and shows that every extremal graph of order $n\ge n_0$ is a blow-up of $B$. We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results., Comment: 44 pages; incorporates reviewers' suggestions
- Published
- 2019
38. Graph of graphs analysis for multiplexed data with application to imaging mass cytometry
- Author
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Ya-Wei Eileen Lin, Kurt A. Schalper, Ronen Talmon, Franz Villarroel-Espindola, Yuval Kluger, Shruti Desai, and Tal Shnitzer
- Subjects
0301 basic medicine ,Lung Neoplasms ,Databases, Factual ,Computer science ,Gaussian ,Cancer Treatment ,02 engineering and technology ,Mass Spectrometry ,Machine Learning ,Mathematical and Statistical Techniques ,Spatial reference system ,0202 electrical engineering, electronic engineering, information engineering ,Medicine and Health Sciences ,Biology (General) ,Image Cytometry ,0303 health sciences ,Ecology ,Covariance ,Mathematical Models ,Applied Mathematics ,Simulation and Modeling ,Estimator ,Random walk ,Graph ,Molecular Imaging ,Computational Theory and Mathematics ,Oncology ,Modeling and Simulation ,Physical Sciences ,symbols ,Probability distribution ,020201 artificial intelligence & image processing ,Algorithms ,Research Article ,medicine.medical_specialty ,QH301-705.5 ,Imaging Techniques ,Feature vector ,Antineoplastic Agents ,Research and Analysis Methods ,symbols.namesake ,03 medical and health sciences ,Cellular and Molecular Neuroscience ,Image Interpretation, Computer-Assisted ,medicine ,Genetics ,Maximum a posteriori estimation ,Humans ,Sensitivity (control systems) ,Representation (mathematics) ,Molecular Biology ,Ecology, Evolution, Behavior and Systematics ,Eigenvalues and eigenvectors ,030304 developmental biology ,business.industry ,Nonlinear dimensionality reduction ,Pattern recognition ,Eigenvalues ,Random Variables ,Data structure ,Probability Theory ,Probability Distribution ,Spectral imaging ,030104 developmental biology ,Algebra ,Linear Algebra ,Random Walk ,Artificial intelligence ,business ,Eigenvectors ,Mathematics - Abstract
Imaging Mass Cytometry (IMC) combines laser ablation and mass spectrometry to quantitate metal-conjugated primary antibodies incubated in intact tumor tissue slides. This strategy allows spatially-resolved multiplexing of dozens of simultaneous protein targets with 1μm resolution. Each slide is a spatial assay consisting of high-dimensional multivariate observations (m-dimensional feature space) collected at different spatial positions and capturing data from a single biological sample or even representative spots from multiple samples when using tissue microarrays. Often, each of these spatial assays could be characterized by several regions of interest (ROIs). To extract meaningful information from the multi-dimensional observations recorded at different ROIs across different assays, we propose to analyze such datasets using a two-step graph-based approach. We first construct for each ROI a graph representing the interactions between the m covariates and compute an m dimensional vector characterizing the steady state distribution among features. We then use all these m-dimensional vectors to construct a graph between the ROIs from all assays. This second graph is subjected to a nonlinear dimension reduction analysis, retrieving the intrinsic geometric representation of the ROIs. Such a representation provides the foundation for efficient and accurate organization of the different ROIs that correlates with their phenotypes. Theoretically, we show that when the ROIs have a particular bi-modal distribution, the new representation gives rise to a better distinction between the two modalities compared to the maximum a posteriori (MAP) estimator. We applied our method to predict the sensitivity to PD-1 axis blockers treatment of lung cancer subjects based on IMC data, achieving 97.3% average accuracy on two IMC datasets. This serves as empirical evidence that the graph of graphs approach enables us to integrate multiple ROIs and the intra-relationships between the features at each ROI, giving rise to an informative representation that is strongly associated with the phenotypic state of the entire image., Author summary We propose a two-step graph-based analyses for high-dimensional multiplexed datasets characterizing ROIs and their inter-relationships. The first step consists of extracting the steady state distribution of the random walk on the graph, which captures the mutual relations between the covariates of each ROI. The second step employs a nonlinear dimensionality reduction on the steady state distributions to construct a map that unravels the intrinsic geometric structure of the ROIs. We theoretically show that when the ROIs have a two-class structure, our method accentuates the distinction between the classes. Particularly, in a setting with Gaussian distribution it outperforms the MAP estimator, implying that the mutual relations between the covariates within the ROIs and spatial coordinates are well captured by the steady state distributions. We apply our method to imaging mass cytometry (IMC). Our analysis provides a representation that facilitates prediction of the sensitivity to PD-1 axis blockers treatment of lung cancer subjects. Particularly, our approach achieves state of the art results with average accuracy of 97.3% on two IMC datasets.
- Published
- 2021
39. Automorphisms groups of simplicial complexes of infinite type surfaces
- Author
-
Jesús Hernández Hernández and Ferrán Valdez
- Subjects
Statistics::Theory ,General Mathematics ,infinite type surface ,010102 general mathematics ,Infinite type surface ,Curve complex ,Automorphism ,01 natural sciences ,Graph ,Mapping class group ,010101 applied mathematics ,Combinatorics ,Algebra ,Mathematics::Group Theory ,Mathematics::Probability ,20F65 ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
Let $S$ be an orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex $\mathcal{C}(S)$, the nonseparating curve complex $\mathcal{N}(S)$, and the Schmutz graph $\mathcal{G}(S)$ of $S$. When all topological ends of $S$ carry genus, we show that all elements in the automorphism groups $\operatorname{Aut}(\mathcal{C}(S))$, $\operatorname{Aut}(\mathcal{N}(S))$, and $\operatorname{Aut}(\mathcal{G}(S))$ are geometric, i.e., these groups are naturally isomorphic to the extended mapping class group $\operatorname{MCG}^{*}(S)$ of the infinite surface $S$. Finally, we study rigidity phenomena within $\operatorname{Aut}(\mathcal{C}(S))$ and $\operatorname{Aut}(\mathcal{N}(S))$.
- Published
- 2021
40. The automorphism group of a graphon.
- Author
-
Lovász, László and Szegedy, Balázs
- Subjects
- *
AUTOMORPHISM groups , *GRAPH theory , *MATHEMATICAL proofs , *ALGEBRA , *MATHEMATICAL models - Abstract
We study the automorphism group of graphons (graph limits). We prove that after an appropriate “standardization” of the graphon, the automorphism group is compact. Furthermore, we characterize the orbits of the automorphism group on k -tuples of points. Among applications we study the graph algebras defined by finite rank graphons and the space of node-transitive graphons. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
41. A methodology and theoretical taxonomy for centrality measures: What are the best centrality indicators for student networks?
- Author
-
Kristel Vignery and Wim Laurier
- Subjects
Computer science ,Social Sciences ,02 engineering and technology ,Surveys ,BIOLOGICAL NETWORKS ,computer.software_genre ,Geodesics ,Infographics ,Mathematical and Statistical Techniques ,Sociology ,Centrality ,TOPOLOGY ,Data Management ,COMPLEX NETWORKS ,Principal Component Analysis ,Multidisciplinary ,05 social sciences ,Statistics ,050301 education ,Classification ,Graph ,Multidisciplinary Sciences ,Social Networks ,Research Design ,Physical Sciences ,Medicine ,Science & Technology - Other Topics ,Graphs ,Network Analysis ,Research Article ,Computer and Information Sciences ,Science ,0206 medical engineering ,Closeness ,POWER ,Geometry ,Machine learning ,Research and Analysis Methods ,Betweenness centrality ,MISSING DATA ,Taxonomy (general) ,Humans ,DISTRIBUTIONS ,Statistical Methods ,Students ,Taxonomy ,Survey Research ,Science & Technology ,IDENTIFICATION ,business.industry ,Data Visualization ,Biology and Life Sciences ,Social Support ,Models, Theoretical ,PERFORMANCE ,FRAMEWORK ,Algebra ,Linear Algebra ,Multivariate Analysis ,Artificial intelligence ,SOCIAL NETWORKS ,business ,Eigenvectors ,0503 education ,computer ,020602 bioinformatics ,Mathematics - Abstract
In order to understand and represent the importance of nodes within networks better, most of the studies that investigate graphs compute the nodes' centrality within their network(s) of interest. In the literature, the most frequent measures used are degree, closeness and/or betweenness centrality, even if other measures might be valid candidates for representing the importance of nodes within networks. The main contribution of this paper is the development of a methodology that allows one to understand, compare and validate centrality indices when studying a particular network of interest. The proposed methodology integrates the following steps: choosing the centrality measures for the network of interest; developing a theoretical taxonomy of these measures; identifying, by means of Principal Component Analysis (PCA), latent dimensions of centrality within the network of interest; verifying the proposed taxonomy of centrality measures; and identifying the centrality measures that best represent the network of interest. Also, we applied the proposed methodology to an existing graph of interest, in our case a real friendship student network. We chose eighteen centrality measures that were developed in SNA and are available and computed in a specific library (CINNA), defined them thoroughly, and proposed a theoretical taxonomy of these eighteen measures. PCA showed the emergence of six latent dimensions of centrality within the student network and saturation of most of the centrality indices on the same categories as those proposed by the theoretical taxonomy. Additionally, the results suggest that indices other than the ones most frequently applied might be more relevant for research on friendship student networks. Finally, the integrated methodology that we propose can be applied to other centrality indices and/or other network types than student graphs. ispartof: PLOS ONE vol:15 issue:12 ispartof: location:United States status: published
- Published
- 2020
42. Lattices from graph associahedra and subalgebras of the Malvenuto–Reutenauer algebra
- Author
-
Emily Barnard and Thomas McConville
- Subjects
Algebra and Number Theory ,Binary tree ,Mathematics::Rings and Algebras ,010102 general mathematics ,0102 computer and information sciences ,Hopf algebra ,01 natural sciences ,Graph ,Algebra ,010201 computation theory & mathematics ,Mathematics::Quantum Algebra ,Lattice (order) ,0101 mathematics ,Partially ordered set ,Quotient ,Mathematics - Abstract
The Malvenuto–Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto–Reutenauer algebra, both of which include the Loday–Ronco algebra. The first is a construction by Reading defined in terms of lattice quotients of the weak order, and the second is a construction by Ronco in terms of graph associahedra. To make this comparison, we consider a natural partial ordering on the maximal tubings of a graph and characterize those graphs for which this poset is a lattice quotient of the weak order.
- Published
- 2020
43. Kemeny-based testing for COVID-19
- Author
-
Ekaterina Dudkina, Robert Shorten, Roderick Murray-Smith, Thomas Parisini, Lewi Stone, Michelangelo Bin, Emanuele Crisostomi, Serife Yilmaz, Pietro Ferraro, Yilmaz, S., Dudkina, E., Bin, M., Crisostomi, E., Ferraro, P., Murray-Smith, R., Parisini, T., Stone, L., and Shorten, R.
- Subjects
0209 industrial biotechnology ,Viral Diseases ,Theoretical computer science ,Computer science ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Infographics ,contact tracing ,020901 industrial engineering & automation ,Mathematical and Statistical Techniques ,Medical Conditions ,COVID-19 Testing ,Medicine and Health Sciences ,Data Management ,Virus Testing ,Multidisciplinary ,Ecology ,Mathematical Models ,Community structure ,Software Engineering ,Graph ,Infectious Diseases ,Community Ecology ,Physical Sciences ,Medicine ,Engineering and Technology ,Coronavirus Infections ,COVID-19 ,networks ,Graphs ,Algorithms ,Research Article ,Physics - Physics and Society ,Computer and Information Sciences ,Markov Models ,Coronavirus disease 2019 (COVID-19) ,Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) ,Science ,Pneumonia, Viral ,FOS: Physical sciences ,Physics and Society (physics.soc-ph) ,Research and Analysis Methods ,Computer Software ,Betacoronavirus ,Diagnostic Medicine ,Humans ,0101 mathematics ,Quantitative Biology - Populations and Evolution ,Community Structure ,Pandemics ,Clinical Laboratory Techniques ,SARS-CoV-2 ,Data Visualization ,Ecology and Environmental Sciences ,Populations and Evolution (q-bio.PE) ,Biology and Life Sciences ,Eigenvalues ,Covid 19 ,Apps ,Models, Theoretical ,Probability Theory ,Algebra ,Linear Algebra ,FOS: Biological sciences ,Random Walk ,network ,Contact Tracing ,Mathematics - Abstract
Testing, tracking and tracing abilities have been identified as pivotal in helping countries to safely reopen activities after the first wave of the COVID-19 virus. Contact tracing apps give the unprecedented possibility to reconstruct graphs of daily contacts, so the question is: who should be tested? As human contact networks are known to exhibit community structure, in this paper we show that the Kemeny constant of a graph can be used to identify and analyze bridges between communities in a graph. Our ‘Kemeny indicator’ is the value of the Kemeny constant in the new graph that is obtained when a node is removed from the original graph. We show that testing individuals who are associated with large values of the Kemeny indicator can help in efficiently intercepting new virus outbreaks, when they are still in their early stage. Extensive simulations provide promising results in early identification and in blocking the possible ‘super-spreaders’ links that transmit disease between different communities.
- Published
- 2020
44. Combinatorial dimensions: Indecomposability on certain local finite-dimensional trivial extension algebras.
- Author
-
Orendain, Juan
- Subjects
- *
COMBINATORICS , *DIMENSIONS , *MATHEMATICAL decomposition , *DIMENSIONAL analysis , *ALGEBRA , *MODULES (Algebra) , *GRAPH theory - Abstract
We study problems related to indecomposability of modules over certain local finite-dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial dimension, and of fundamental combinatorial dimension of a module. We use these concepts to establish, under favorable conditions, criteria for the indecomposability of a module. We present categorified versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
45. Isomorphy and dilatation in digraphs.
- Author
-
Dammak, Jamel
- Subjects
- *
DIRECTED graphs , *GRAPH theory , *APPLIED mathematics , *ALGEBRA , *MATHEMATICAL analysis - Abstract
Given a digraph , or more simply ( V, A). Two digraphs G and G' are hemimorphic if G' is isomorphic to G or to its dual . For , denote and . Given two digraphs G and H with the property that , let . We say that we dilate v0 by H if we replace v0 by H, obtaining then a new digraph R satisfying: for all and for all is and . We say that the digraph R is obtained from G by dilating the vertex v0 by the digraph H. Our main result is: let σ be an isomorphism from a digraph M onto a digraph M' and such that and H, H' be two digraphs. Consider the digraph R (resp. R') obtained by dilating v0 (resp. ) in M (resp. in M') by H (resp. H'). If R and R' are hemimorphic and H and H' are hemimorphic, then R and R' are isomorphic and H and H' are isomorphic. The main result generalizes those of Y. Boudabbous and J. Dammak and of M. Bouaziz and Y. Boudabbous. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
46. Application of the Handshaking Lemma in the Dyeing Theory of Graph
- Author
-
Daniel Ofori-Kusi, Richmond Nii Okle, Elise Hamunyela, Roland Forson, and Guanghui Cai
- Subjects
Basis (linear algebra) ,Computer science ,Handshaking lemma ,Topology (electrical circuits) ,Graph theory ,GeneralLiterature_MISCELLANEOUS ,Graph ,Algebra ,Transfer (group theory) ,symbols.namesake ,Development (topology) ,Euler's formula ,symbols ,Graph (abstract data type) ,Dyeing - Abstract
The handshaking lemma is one of the important branches of graph theory. The content is widely applied in topology and computer science. The basis of the development of the dyeing theory used in this research paper is to discuss the application of the right transfer method in dyeing theory.
- Published
- 2020
47. An Alternative Reformulation of the Transformation Rules in the Beta Part of Peirce’s Existential Graphs
- Author
-
Shigeyuki Atarashi
- Subjects
Algebra ,Diagrammatic reasoning ,Thick line ,Philosophy ,Presumption ,Domain of discourse ,Graph ,Thin line ,Existentialism - Abstract
The aim of this paper is to reformulate the transformation rules presented by Peirce in the Beta part of Existential Graphs, in a different way from the rules systemized by Roberts and Shin. Existential Graphs provides an iconic system of logic. In other words, it visualizes logical reasonings by using diagrammatic representations. Specifically, a graph represents a situation occurring in a certain universe of discourse. In addition, Peirce introduced a line of identity and a cut. The former is a thick line that affirms the identity of two particulars signified by its two ends. The latter is a closed curve that is drawn with a thin line. By enclosing a graph entirely by a cut, the content represented by the graph is denied. Peirce forbid a line of identity from crossing a cut, yet both Roberts and Shin presumed that a line of identity can cross a cut. Hence, this paper eliminates that presumption completely and shows an alternative reformulation of the transformation rules in the Beta part of Existential Graphs.
- Published
- 2020
48. The braid index of DNA double crossover polyhedral links
- Author
-
Yuanan Diao and Xiao-Sheng Cheng
- Subjects
Models, Molecular ,Computer and Information Sciences ,Molecular biology ,Science ,Mathematical properties ,Plant Science ,010402 general chemistry ,01 natural sciences ,Infographics ,Polynomials ,Biochemistry ,Topology ,Combinatorics ,Polyhedron ,Knot Theory ,Braid ,Genetics ,Flower Anatomy ,Mathematics ,Multidisciplinary ,010405 organic chemistry ,Data Visualization ,Plant Anatomy ,Double crossover ,Chemical Compounds ,Biology and Life Sciences ,DNA structure ,Graph theory ,DNA ,Graph ,Hydrocarbons ,0104 chemical sciences ,Knot theory ,Nucleic acids ,Macromolecular structure analysis ,Chemistry ,Petals ,Algebra ,Graph Theory ,Physical Sciences ,Nucleic Acid Conformation ,Medicine ,Graphs ,Research Article - Abstract
In this paper, the authors study the mathematical properties of a class of alternating links called polyhedral links which have been used to model DNA polyhedra. The motivation of such studies is to provide guidance and aid in the research of the properties of certain DNA molecules. For example, such studies can provide characterizations of the structural complexity of DNA molecules. In an earlier work, Cheng and Jin studied the mathematical properties of such polyhedral links and were able to determine the braid index of a double crossover polyhedral link with 4 turn. However, the braid index of a double crossover polyhedral link with 4.5 turn remained an unsolved problem to this date, even though the graphs that admit the double crossover polyhedral links with 4.5 turn have been synthesized. In this paper, we provide a complete formulation of the braid index of a double crossover polyhedral link with an arbitrary turn number. Our approach is more general and it allows us to completely determine the braid indices for a much larger class of links. In the case of the double crossover polyhedral links, our formulation of the braid index is a simple formula based on a simpler graph used as a template to build the double crossover polyhedral links.
- Published
- 2020
49. The Linear Algebra of Graphs
- Author
-
Charu C. Aggarwal
- Subjects
Algebra ,Algebraic properties ,Optimization problem ,Computer science ,Linear algebra ,Telecommunications network ,Graph ,Spectral clustering - Abstract
Graphs are encountered in many real-world settings, such as the Web, social networks, and communication networks. Furthermore, many machine learning applications are conceptually represented as optimization problems on graphs. Graph matrices have a number of useful algebraic properties, which can be leveraged in machine learning. There are close connections between kernels and the linear algebra of graphs; a classical application that naturally belongs to both fields is spectral clustering (cf. Section 10.5).
- Published
- 2020
50. Homotopy classification of homogeneous projections in the corona algebra of C(X,B) for a graph X
- Author
-
Hyun Ho Lee
- Subjects
Simple graph ,Applied Mathematics ,Homotopy ,010102 general mathematics ,Directed graph ,01 natural sciences ,Graph ,Algebra ,Homogeneous ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper we consider certain projections in the corona algebra of C ( X ) ⊗ B for X a directed graph and B a stable C ⁎ -algebra of which the multiplier algebra has real rank zero. Assuming a kind of homogeneity on the projections we characterize when two such projections are homotopy equivalent. As a result, for a simple graph we show that such projections are always homotopic.
- Published
- 2018
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