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Let G be a connected graph, the sum geometric - arithmetic means index of G is defined as SGAM (G) = ∑uv ∈E(G) (√(d(u)d(v) )+ (d(v)+d(v))/ 2 ) . In this paper, the concept of sum geometric harmonic means index of a graph G, denoted by SG hm (G) is introduced and SGhm (G) of few families of graphs is computed. Further, we establish the bounds for sum geometric harmonic means index. General Terms: AMS, subject Classification 05C07; 92E10. [ABSTRACT FROM AUTHOR]

We define, for any graph G = (V , E) , a boundary ∂ G ⊆ V. The definition coincides with what one would expected for the discretization of (sufficiently nice) Euclidean domains and contains all vertices from the Chartrand, Erwin, Johns & Zhang boundary. Moreover, it satisfies an isoperimetric principle stating that graphs with many vertices have a large boundary unless they contain long paths: we show that for graphs with maximaldegree Δ | ∂ G | ≥ 1 2 Δ | V | diam (G) . For graphs discretizing Euclidean domains, one has diam (G) ∼ | V | 1 / d and recovers the scaling of the classical Euclidean isoperimetric principle. [ABSTRACT FROM AUTHOR]

The emergence of image-based systems to improve diagnostic pathology precision, involving the intent to label sets or bags of instances, greatly hinges on Multiple Instance Learning for Whole Slide Images(WSIs). Contemporary works have shown excellent performance for a neural network in MIL settings. Here, we examine a graph-based model to facilitate end-to-end learning and sample suitable patches using a tile-based approach. We propose MIL-GNN to employ a graph-based Variational Auto-encoder with a Gaussian mixture model to discover relations between sample patches for the purposes to aggregate patch details into an individual vector representation. Using the classical MIL dataset MUSK and distinguishing two lung cancer sub-types, lung cancer called adenocarcinoma (LUAD) and lung squamous cell carcinoma (LUSC), we exhibit the efficacy of our technique. We achieved a 97.42% accuracy on the MUSK dataset and a 94.3% AUC on the classification of lung cancer sub-types utilizing features. [ABSTRACT FROM AUTHOR]

An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR-structures were introduced in a paper by P. Potočnik and S. Wilson (2014) [12] , as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper. [ABSTRACT FROM AUTHOR]

A graph is a cograph if and only if it has no induced path on 4 vertices. The twin reduction graph of a graph G is denoted by R G. In this article, we describe the Laplacian eigenvalues and eigenvectors of a cograph G using its twin reduction graph R G , and characterize cographs with even and odd integer eigenvalues, respectively. We provide a construction for the Laplacian cospectral cographs. Further, we provide a complete description of the Laplacian spectrum of H -join of graphs when H is a cograph, and then obtain bounds for the algebraic connectivity of such graphs. We also provide some interesting observations on Hadamard diagonalizable cographs. [ABSTRACT FROM AUTHOR]

In 1975, P. Erdős proposed the problem of determining the maximum number f (n) of edges in a graph on n vertices in which any two cycles are of different lengths. Let f ∗ (n) be the maximum number of edges in a simple graph on n vertices in which any two cycles are of different lengths. Let M n be the set of simple graphs on n vertices and f ∗ (n) edges in which any two cycles are of different lengths. Let m c (n) be the maximum cycle length for all G ∈ M n . In this paper, it is proved that for n sufficiently large, m c (n) ≤ 15 16 n. We make the following conjecture: Conjecture. lim n → ∞ m c (n) n = 0. [ABSTRACT FROM AUTHOR]

Blocking flows were introduced by Dinic (Soviet Math Doklady 11: 1277–1280, 1970) to speed up the computation of maximum network flows. They have been used in algorithms for problems such as maximum cardinality matching of bipartite graphs Hopcroft and Karp (SIAM J Comput 2(4), 225–231, 1973) and general graphs Micali and Vazirani (in: Proceedings of the 21st Annual Symposium on Foundations of Computer Science, 17–27, 1980), maximum weight matching of general graphs Gabow and Tarjan (J ACM 38(4), 815–853, 1991), and many others. The blocking algorithm of Gabow and Tarjan (1991) for matching is based on depth-first search. We extend the depth-first search approach to find f-factors of general multigraphs. Here f is an arbitrary integral-valued function on vertices, an f-matching is a subgraph where every vertex x has degree ≤ f (x) , an f-factor has equality in every degree bound. A set of blocking trails for an f-matching M is a maximal collection A of edge-disjoint augmenting trails such that M ⨁ A ∈ A A is a valid f-matching. Blocking trails are needed in efficient algorithms for maximum cardinality f-matching Huang and Pettie (Algorithmica 84(7): 1952–1992, 2022), maximum weight f-factors/matchings by scaling Duan et al. (In: Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), Vol. 168 of LIPIcs, 41:1-41:17, 2020; Gabow (A weight-scaling algorithm for f-factors of multigraphs.arXiv:2010.01102, 2020), and approximate maximum weight f-factors and f-edge covers Huang and Pettie (2022). Since these algorithms find many sets of blocking trails, the time to find blocking trails is a dominant factor in the running time. Our blocking trail algorithm runs in linear time O(m). In independent work and using a different approach, Huang and Pettie (2022) present a blocking trails algorithm using time O (m α (m , n)) . As examples of the time bounds for the above applications, an approximate maximum weight f-factor is found in time O (m α (m , n)) using Huang and Pettie (2022), and our algorithm eliminates the factor α (m , n) . Similarly a maximum weight f-factor is found in time O (Φ log Φ m α (m , n) log (Φ W)) using Huang and Pettie (2022) , ( Φ = ∑ v ∈ V f (v) , W the maximum edge weight) and our algorithm eliminates the α (m , n) factor, making the time within a factor log Φ of the bound for bipartite multigraphs. The technical difficulty for this work stems from the fact that a fixed vertex can occur many times in a given search. This does not occur in ordinary matching or in algorithms for maximum cardinality or maximum weight f-matching. These multiple occurrences can create a new variant of blossom, the "skew blossom". Also they can make blossoms become "incomplete", i.e., partially processed yet still relevant in future searches. [ABSTRACT FROM AUTHOR]

The challenge for graph matching algorithms is to extend known time bounds for bipartite graphs to general graphs. We discuss combinatorial algorithms for finding a maximum weight f-factor on an arbitrary multigraph, for given integral weights of magnitude at most W. (An f-factor is a subgraph whose degree function is the given function f : V → N .) For simple bipartite graphs the best-known time bound for combinatorial algorithms is O (n 2 / 3 m log n W) [Gabow and Tarjan, SIAM J Comput 18(5):1013–1036, 1989; n and m are respectively the number of vertices and edges.] A recent algorithm of Duan et al. [in: Proc. of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), 2020] for f-factors of simple general graphs comes within logarithmic factors of this bound, O ~ (n 2 / 3 m log W) . The best-known bound for bipartite multigraphs is O (Φ m log Φ W) ( Φ ≤ m is the size of the f-factor, Φ = ∑ v ∈ V f (v) / 2 ). This bound is more general than the restriction to simple graphs, and is even superior on "small" simple graphs, i.e., Φ = o (n 4 / 3) . We present an algorithm that comes within a log Φ factor of this bound, i.e., O (Φ log Φ m log Φ W) . The algorithm is a direct generalization of the algorithm of Gabow and Tarjan [J ACM 38(4):815–853, 1991] for the special case of ordinary matching ( f ≡ 1 ). We present that algorithm first. Our analysis is a simplified and more concrete version of Gabow and Tarjan [J ACM 38(4):815–853, 1991] and has independent interest. Furthermore the algorithm and analysis are both incorporated, without modification, into the f-factor algorithm. To extend these ideas to f-factors, the first step is "expanding" edges (i.e., replacing an edge by a length 3 alternating path). Duan et al. [in: Proc. of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), 2020] uses a one-time expansion of the entire graph. In contrast, our algorithm keeps the graph small by only expanding selected edges (edges incident to blossoms, in "I(B) sets"). Expanded edges get "compressed" back to their source when no longer needed. Expansion necessitates using an alternate graph model for blossoms (we call them "e-blossoms"). Compression requires coordinating e-blossoms with standard blossoms. [ABSTRACT FROM AUTHOR]

In this paper, we consider the graphs that have a unique set that achieves a domination parameter and determine the maximum value of the associated parameter, both in graphs in general and in graphs without end-vertices. For example, for a graph G of order n without an end-vertex, we show an upper bound of n/3 if G has a unique minimum dominating set, an upper bound of (n - 1) / 2 if G has a unique minimum total dominating set or a unique minimum 2-dominating set, and an upper bound of n/2 if G has a unique paired dominating set. [ABSTRACT FROM AUTHOR]

In this work we show that if iterates of a nonexpansive self-mapping of a complete metric with a graph converge uniformly on a subset of the space, then this convergence is stable under the presence of small computational errors. [ABSTRACT FROM AUTHOR]

The author’s recent research papers, “Cumulative deviation of a subpopulation from the full population” and “A graphical method of cumulative differences between two subpopulations” (both published in volume 8 of Springer’s open-access Journal of Big Data during 2021), propose graphical methods and summary statistics, without extensively calibrating formal significance tests. The summary metrics and methods can measure the calibration of probabilistic predictions and can assess differences in responses between a subpopulation and the full population while controlling for a covariate or score via conditioning on it. These recently published papers construct significance tests based on the scalar summary statistics, but only sketch how to calibrate the attained significance levels (also known as “P-values”) for the tests. The present article reviews and synthesizes work spanning many decades in order to detail how to calibrate the P-values. The present paper presents computationally efficient, easily implemented numerical methods for evaluating properly calibrated P-values, together with rigorous mathematical proofs guaranteeing their accuracy, and illustrates and validates the methods with open-source software and numerical examples. [ABSTRACT FROM AUTHOR]

In our joint work with Dan Butnariu (2006) we established the stability of the convergence of iterates of a nonexpansive mapping on a complete metric space in the presence of summable computational errors. In a recent paper of ours, we extended this result to inexact iterates of nonexpansive mappings on complete metric spaces with graphs under a certain assumption on the iterates. In the present paper we obtain an analogous result by removing that assumption on the iterates and replacing it with an additional assumption on the graph. [ABSTRACT FROM AUTHOR]

We consider the problem of finding an accurate representation of neuron shapes, extracting sub-cellular features, and classifying neurons based on neuron shapes. In neuroscience research, the skeleton representation is often used as a compact and abstract representation of neuron shapes. However, existing methods are limited to getting and analyzing "curve" skeletons which can only be applied for tubular shapes. This paper presents a 3D neuron morphology analysis method for more general and complex neuron shapes. First, we introduce the concept of skeleton mesh to represent general neuron shapes and propose a novel method for computing mesh representations from 3D surface point clouds. A skeleton graph is then obtained from skeleton mesh and is used to extract sub-cellular features. Finally, an unsupervised learning method is used to embed the skeleton graph for neuron classification. Extensive experiment results are provided and demonstrate the robustness of our method to analyze neuron morphology. [ABSTRACT FROM AUTHOR]

We describe and evaluate a method to mitigate unwanted student collaborations in assessments, which we recently implemented in a second-year undergraduate mathematics module. The method requires a list of specific pairs of students to be prevented from collaborating, which we constructed based on the results of previous assessments. We converted the list into an undirected graph – in which vertices represent students and adjacent vertices represent pairs of students to be prevented from collaborating – and determined a proper colouring of this graph. Subsequently, we designed multi-version assessments, the number and the distribution of the versions being as prescribed by the proper colouring. To assess this method's effectiveness, we conducted an anonymous survey collecting the students' opinions. The results showed that the method indeed reduced collaborations, and that it did so more significantly in the case of quizzes than take-home assignments. [ABSTRACT FROM AUTHOR]

Mixed-Integer Linear Programs (MILPs) have many practical applications. Most modelling tools for MILPs fall in two broad categories. Indeed, tools such as algebraic modelling languages allow practitioners to compactly encode models using syntax close to mathematical notation but usually lack support for special structures, while other tools instead provide predefined components that can be easily assembled but modifying or adding new components is difficult. In this work, we present the inner workings of the Graph-Based Optimization Modelling Language (GBOML), an open-source modelling tool implemented in Python combining the strengths of both worlds. GBOML natively supports special structures that can be encoded by a hierarchical hypergraph, offers syntax close to mathematical notation and facilitates the modular construction and reuse of time-indexed models. We detail design choices enabling these features and show that they simplify problem encoding, lead to faster instance generation times and sometimes faster solve times. We benchmark the times taken by GBOML, JuMP, Plasmo, Pyomo and AMPL to generate instances of a structured MILP. We find that GBOML outperforms Plasmo and Pyomo, is tied with JuMP but is slower than AMPL. With parallel model generation, GBOML outperforms JuMP and closes the gap with AMPL. GBOML has the smallest memory footprint.  [ABSTRACT FROM AUTHOR]

Social networking is prevalent everywhere. Technology has made the world seem smaller by bringing individuals together, reconnecting people, and providing other opportunities for connection. However, the risk of data fraud has also increased. The early detection of fraud needs a deep analysis of tweets, which has been done using Neo4J. The Twitter API receives requests from thousands of developers every single day. Limits are imposed on the number of requests that may be made to assist in managing the overwhelming amount of these requests. The most typical request limit period is 15 min. If an endpoint has a rate limit of 1000 requests/15 min, then 1000 requests are permitted over 15 min. Every time a real-time tweet analysis error is returned to the application when these limits are exceeded, online updation of the Neo4J database is difficult. In the proposed method, first and foremost, a new time-based cache (TmCache) is introduced between the database and Twitter API, which removes the complexities of Twitter API and reduces the analyzing time by 85.36%. Furthermore, a machine learning-based approach Multinomial Naive Bayes classifier is used to classify the spam and non-spam tweets with the lowest training time and 96.6% accuracy. The proposed model is helpful in the early detection of spam accurately and can be used in reducing the risk of propagation of agendas which are a major threat to society at large in the current era of social networking. [ABSTRACT FROM AUTHOR]

The quality of the bicycle network determines ridership, safety, connectivity, equity, and livability. Very few former research studies investigated network connectivity for individual user types and identify network needs and barriers based on these rider types. This study measures the network connectivity for different rider types using level of traffic stress (LTS) and graph theory concepts. As a symbolic representation of a road network and its connectivity, a graph represents the structural properties of networks and compares one measure over another by taking into account spatial features. In addition, this study defines a bicycle network for different types of riders using LTS metrics based on traffic speed, road geometry, and traffic volume. This study evaluates the OpenStreetMap (OSM) bicycle network for Portland, Oregon, as a case study. Three transit stations in the downtown, riverside, and residential area were considered to assess the connectivity and barriers with a home at block level for last and first-mile coverage. The analysis shows that 29% of links in Portland need to be improved with more bicycle facilities to provide access to basic adult riders, and 33% of links require improvement for children. The networks are well connected for “strong and fearless” and “confident and enthused” users but not well connected for basic adults and children in many neighborhoods with low alpha and grid tree pattern (GTP) indices. The results indicate that planners and designers need to improve their network connectivity for all types of users to ensure equal active transportation opportunities beyond a particular portion of the network. [ABSTRACT FROM AUTHOR]

In 2007, in our joint work with D. Butnariu and S. Reich, we proved that if for a self-mapping of a complete metric that is uniformly continuous on bounded sets all its iterates converge uniformly on bounded sets, then this convergence is stable under the presence of small errors. In the present paper, we obtain an extension of this result for self-mappings of a metric space with a graph. [ABSTRACT FROM AUTHOR]

The matroid concept is an important model in real life applications. Determining the existence of mutations of DNA and RNA plays an essential role in biological studies. The matroidal structures of matrices are used for determining the existence of mutations of DNA; graph theory and matroid theory can be used to identify important mutations in genetic data. We construct an algorithm to determine the existence of a mutation. Finally, we study the similarity and dissimilarity between genes using matroids. [ABSTRACT FROM AUTHOR]

We consider the evolution of phylogenetic gene trees along phylogenetic species networks, according to the network multispecies coalescent process, and introduce a new network coalescent model with correlated inheritance of gene flow. This model generalizes two traditional versions of the network coalescent: with independent or common inheritance. At each reticulation, multiple lineages of a given locus are inherited from parental populations chosen at random, either independently across lineages or with positive correlation according to a Dirichlet process. This process may account for locus-specific probabilities of inheritance, for example. We implemented the simulation of gene trees under these network coalescent models in the Julia package PhyloCoalSimulations, which depends on PhyloNetworks and its powerful network manipulation tools. Input species phylogenies can be read in extended Newick format, either in numbers of generations or in coalescent units. Simulated gene trees can be written in Newick format, and in a way that preserves information about their embedding within the species network. This embedding can be used for downstream purposes, such as to simulate species-specific processes like rate variation across species, or for other scenarios as illustrated in this note. This package should be useful for simulation studies and simulation-based inference methods. The software is available open source with documentation and a tutorial at https://github.com/cecileane/PhyloCoalSimulations.jl. [ABSTRACT FROM AUTHOR]

For q ∈ R , the Q - matrix Q = Q q of a connected simple graph G = (V , E) is Q q = (q ∂ (x , y)) x , y ∈ V , where ∂ denotes the path-length distance. Describing the set π (G) consisting of those q ∈ R for which Q q is positive semidefinite is fundamental in asymptotic spectral analysis of graphs from the viewpoint of quantum probability theory. Assume that G has at least two vertices. Then π (G) is easily seen to be a nonempty closed subset of the interval [ − 1 , 1 ]. In this note, we show that π (G) = [ − 1 , 1 ] if and only if G is isometrically embeddable into a hypercube (infinite-dimensional if G is infinite) if and only if G is bipartite and does not possess certain five-vertex configurations, an example of which is an induced K 2 , 3. [ABSTRACT FROM AUTHOR]

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]

We present the isomorphic classification of free topological groups over the spaces, which are geometric realizations of finite graphs. [ABSTRACT FROM AUTHOR]

Several molecular mechanisms of thalidomide embryopathy (TE) have been investigated, from anti-angiogenesis to oxidative stress to cereblon binding. Recently, it was discovered that thalidomide and its analogs, named immunomodulatory drugs (IMiDs), induced the degradation of C2H2 transcription factors (TFs). This mechanism might impact the strict transcriptional regulation of the developing embryo. Hence, this study aims to evaluate the TFs altered by IMiDs, prioritizing the ones associated with embryogenesis through transcriptome and systems biology-allied analyses. This study comprises only the experimental data accessed through bioinformatics databases. First, proteins and genes reported in the literature as altered/affected by the IMiDs were annotated. A protein systems biology network was evaluated. TFs beta-catenin (CTNNB1) and SP1 play more central roles: beta-catenin is an essential protein in the network, while SP1 is a putative C2H2 candidate for IMiD-induced degradation. Separately, the differential expressions of the annotated genes were analyzed through 23 publicly available transcriptomes, presenting 8624 differentially expressed genes (2947 in two or more datasets). Seventeen C2H2 TFs were identified as related to embryonic development but not studied for IMiD exposure; these TFs are potential IMiDs degradation neosubstrates. This is the first study to suggest an integration of IMiD molecular mechanisms through C2H2 TF degradation. [ABSTRACT FROM AUTHOR]

This manuscript introduces a finite collection of generalized permutohedra associated to a simple graph. The first polytope of this collection is the graphical zonotope of the graph, and the last is the graph-associahedron associated to it. We describe the weighted integer points enumerators for polytopes in this collection as Hopf algebra morphisms of combinatorial Hopf algebras of decorated graphs. In the last section, we study some properties related to H-polytopes. [ABSTRACT FROM AUTHOR]

This paper presents a way to represent a group using a graph, which involves the concept of a generator element of a group. The graph representing a group is called the generator graph. In the generator graph, the vertices correspond to the elements of the group, and two vertices, x and y, are connected by an edge if either x or y serves as a generator for the group. The paper investigates some properties of these generator graphs and obtains the generator graphs for specific groups. Additionally, it explores the relationship between the generator graph of a group and the generating graph introduced by Lucchini et al. in their work [7]. [ABSTRACT FROM AUTHOR]

Let G be a graph. For a set H of connected graphs, a spanning subgraph H of a graph G is called an H-factor of G if each component of H is isomorphic to a member of H. An H-factor is also referred as a component factor. If G - e admits an H-factor for any e ∈ E(G), then we say that G is an H-factor deleted graph. Let k ≥ 2 be an integer. In this article, we verify that (i) a graph G admits a {K1,1, K1,2, . . ., K1,k, T(2k + 1)}-factor if and only if its binding number bind(G) ≥ ...; (ii) a graph G with δ(G) ≥ 2 is a {K1,1,K1,2, . . ., K1,k, T (2k + 1)}-factor deleted graph if its binding number bind(G) ≥ .... [ABSTRACT FROM AUTHOR]

Modeling data transmission problems in graph theory is internalized to the existence of fractional flows, and thus can be surrogated to be characterized by a fractional factor in diversified settings. We study the fractional factor framework in the network environment when some sites are damaged. The setting we focus on refers to the lower and upper fractional degrees described by two functions on the vertex set. It is determined that G is fractional (g , f , n) critical if δ (G) ≥ ⌊ a 2 + b 2 + 2 a b + 2 a + 2 b − 3 4 a ⌋ + n and I (G) > n + ⌊ (a + b − 1) 2 2 a + 2 b − 1 a ⌋ 2 , where 1 ≤ a ≤ b and b ≥ 2 . [ABSTRACT FROM AUTHOR]

We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by K>0 $K\gt 0$ have diameter bounded by diam(G)≤2∕K $\text{diam}(G)\le 2\unicode{x02215}K$ (a Bonnet–Myers theorem), that diam(G)=2∕K $\text{diam}(G)=2\unicode{x02215}K$ implies that G $G$ has constant curvature (a Cheng theorem) and that there is a spectral gap λ1≥K∕(2n) ${\lambda }_{1}\ge K\unicode{x02215}(2n)$ (a Lichnerowicz theorem). It is computed for several families of graphs and often coincides with Ollivier curvature or Lin–Lu–Yau curvature. The von Neumann Minimax theorem features prominently in the proofs. [ABSTRACT FROM AUTHOR]

The Sombor index of the graph G is a degree based topological index, defined as SO=∑uv∈E(G)d²u+d²v--√, where du is the degree of the vertex u, and E(G) is the edge set of G. Bounds on SO are established in terms of graph energy, size of minimum vertex cover, matching number, and induced matching number. [ABSTRACT FROM AUTHOR]

The graph model is an appreciable tool for data transmission network, where the feasibility of data transmission in site attack circumstances can be described by fractional critical graphs, and the vulnerability of networks can be measured by isolation toughness variant. This paper considers both the stability of the network and the feasibility of data transmission when the sites are destroyed, and determines the isolated toughness variant bound for fractional -critical graphs, where the parameter represents the number of damaged sites at a certain moment. A counterexample proves the sharpness of the given isolated toughness variant bound. The main theoretical conclusion provides an equilibrium between performance and cost in network topology designing. [ABSTRACT FROM AUTHOR]

Inspired by the split decomposition of graphs and rank-width, we introduce the notion of r-splits. We focus on the family of r-splits of a graph of order n, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only O(nr+1) of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least (nr) hyperedges to be represented, using a generalization of set orthogonality. [ABSTRACT FROM AUTHOR]

We describe an algebraic technique for operating with power series whose coefficients are represented by integrals of symmetric functions fn defined on the Cartesian powers Ωn of a set Ω with a measure μ. Moreover, each of the coefficient functions fn is obtained by means of a special mapping from graphs with n labeled vertices belonging to a fixed class. This technique has application to equilibrium statistical mechanics and to problems of enumeration of graphs. [ABSTRACT FROM AUTHOR]

We introduce a unifying approach for invariants of finite matroids that count mappings to a finite set. The aim of this paper is to show that if the cardinalities of mappings with fixed values on a restricted set satisfy contraction–deletion rules, then there is a relation among them that can be expressed in terms of linear algebra. In this way, we study regular chain groups, nowhere-zero flows and tensions on graphs, and acyclic and totally cyclic orientations of oriented matroids and graphs. [ABSTRACT FROM AUTHOR]

Let F t (n) denote the family of all connected graphs of order n with clique number t. In this paper, we present a new upper bound for the chromatic polynomial of a graph G in F t (n) in terms of the clique number of G. Moreover, we also show that the conjecture proposed by Tomescu, which says that if x ≥ k ≥ 4 and G is a connected graph on n vertices with chromatic number k, then P (G , x) ≤ (x) k (x - 1) n - k holds under certain conditions, where (x) k = x (x - 1) ⋯ (x - k + 1) , such as the clique number ω (G) = k ≥ 4 , ω (G) = k - 1 (k ≥ 7) with two (k - 1) -cliques having at most k - 3 vertices in common, or maximum degree ▵ (G) = n - 2 . [ABSTRACT FROM AUTHOR]

This article is an update to Jann (2018a, Stata Journal 18: 765–785). It contains a comprehensive discussion of the colorpalette command, including various changes and additions that have been made to the software since its first publication. Command colorpalette provides colors for use in Stata graphics. In addition to Stata's default colors, colorpalette supports a variety of named colors, a selection of palettes that have been proposed by users, numerous collections of palettes and colormaps from sources such as ColorBrewer, Carto, D3.js, or Matplotlib, as well as color generators in different color spaces. Furthermore, a new command called colorcheck is presented that can be used to evaluate whether colors are distinguishable by people suffering from color vision deficiency. [ABSTRACT FROM AUTHOR]

For a set H of connected graphs, a spanning subgraph H of G is called an H -factor of G if each component of H is isomorphic to an element of H . A graph G is called an H -factor uniform graph if for any two edges e 1 and e 2 of G, G has an H -factor covering e 1 and excluding e 2 . Let each component in H be a path with at least d vertices, where d ≥ 2 is an integer. Then an H -factor and an H -factor uniform graph are called a P ≥ d -factor and a P ≥ d -factor uniform graph, respectively. In this article, we verify that (i) a 2-edge-connected graph G is a P ≥ 3 -factor uniform graph if δ (G) > α (G) + 4 2 ; (ii) a (k + 2) -connected graph G of order n with n ≥ 5 k + 3 - 3 5 γ - 1 is a P ≥ 3 -factor uniform graph if | N G (A) | > γ (n - 3 k - 2) + k + 2 for any independent set A of G with | A | = ⌊ γ (2 k + 1) ⌋ , where k is a positive integer and γ is a real number with 1 3 ≤ γ ≤ 1 . [ABSTRACT FROM AUTHOR]

We study the behavior of the Lorentzian Engle-Pereira-Rovelli-Livine spinfoam amplitude with homogeneous boundary data, under a graph refinement going from five to twenty boundary tetrahedra. This can be interpreted as a wave function of the Universe, for which we compute boundary geometrical operators, correlation functions, and entanglement entropy. The numerical calculation is made possible by adapting the Metropolis-Hastings algorithm, along with recently developed computational methods appropriate for the deep quantum regime. We confirm that the transition amplitudes are stable against such refinement. We find that the average boundary geometry does not change, but the new degrees of freedom correct the quantum fluctuations of the boundary and the correlations between spatial patches. The expectation values are compatible with their geometrical interpretation and the correlations between neighboring patches decay when computed across different spinfoam vertices. [ABSTRACT FROM AUTHOR]

Quantum codes are crucial building blocks of quantum computers. With a self-dual quantum code is attached, canonically, a unique stabilised quantum state. Improving on a previous publication, we show how to determine the coefficients on the basis of kets in these states. Two important ingredients of the proof are algebraic graph theory and quadratic forms. The Arf invariant, in particular, plays a significant role. [ABSTRACT FROM AUTHOR]

Let Heff (Veff, Eeff) be a finite simple connected graph of order m with vertex set Veff and edge set Eeff. A dominating set S d s ⊆ V e f f is called an efficiently dominating set if, for every vertex u a ∈ V G ,   N G u a ∩ S d s = 1 —where NG [ua] denotes the closed neighborhood of the vertex ua. Using efficient domination techniques and labelling, we constructed the fuzzy network. An algorithm has been framed to encrypt and decrypt the secret information present in the network, and furthermore, the algorithm has been given in pseudocode. The mathematical modelling of a strong fuzzy network is defined and constructed to elude the burgeoning intruder. Using the study of the efficient domination of fuzzy graphs, this domination parameter plays a nuanced role in encrypting and decrypting the framed network. One of the main purposes of fuzzy networks is encryption, so one of our contributions to this research is to build a novel combinatorial technique to encrypt and decode the built-in fuzzy network with a secret number utilizing effective domination. An illustration with an appropriate secret message is provided along with the encryption and decryption algorithms. Furthermore, we continued this study in intuitionistic fuzzy networks. [ABSTRACT FROM AUTHOR]

Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph Ω provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first Zagreb index of a graph Ω is defined as the product of the sum of the α th powers of the vertex degrees of Ω , where α is a real number such that α ≠ 0 and α ≠ 1 . The focus of this work is on the extremal graphs for several classes of graphs including trees, unicyclic, and bicyclic graphs, with respect to the generalized multiplicative first Zagreb index. In the initial step, we identify a set of operations that either increases or decreases the generalized multiplicative first Zagreb index for graphs. We then involve analysis of the generalized multiplicative first Zagreb index achieving sharp bounds by characterizing the maximum or minimum graphs for those classes. We present applications of the generalized multiplicative first Zagreb index Π 1 α for predicting the π -electronic energy E π (β) of benzenoid hydrocarbons. In particular, we answer the question concerning the value of α for which the predictive potential of Π 1 α with E π for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that Π 1 α correlates with E π of lower benzenoid hydrocarbons with correlation coefficient ρ = − 0.998 , if α = − 0.00496 . In QSPR modeling, the value ρ = − 0.998 is considered to be considerably significant. [ABSTRACT FROM AUTHOR]

We characterise the weights of all codewords of the hull of the binary code spanned by rows of an incidence matrix Bn of the complete graph Kn for n ≥ 4 and n even. Among others, we show that every codeword of the hull is the sum of an even number of rows of Bn and that its weight depends on the number of rows required to span it. We also determine the weight enumerator and the MacWilliams identity for the hull. [ABSTRACT FROM AUTHOR]

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. In the paper, the crossing number of the join product G* + Dn for the connected graph G* on six vertices consisting of one path on four vertices P4 and two leaves adjacent with the same outer vertex of the path P4 is given, where Dn consists of n isolated vertices. Finally, by adding some edges to the graph G*, we obtain the crossing numbers of the join products of other four graphs with Dn. [ABSTRACT FROM AUTHOR]

Let F t (n) denote the family of all connected graphs of order n with clique number t. In this paper, we present a new upper bound for the chromatic polynomial of a graph G in F t (n) in terms of the clique number of G. Moreover, we also show that the conjecture proposed by Tomescu, which says that if x ≥ k ≥ 4 and G is a connected graph on n vertices with chromatic number k, then P (G , x) ≤ (x) k (x - 1) n - k holds under certain conditions, where (x) k = x (x - 1) ⋯ (x - k + 1) , such as the clique number ω (G) = k ≥ 4 , ω (G) = k - 1 (k ≥ 7) with two (k - 1) -cliques having at most k - 3 vertices in common, or maximum degree ▵ (G) = n - 2 . [ABSTRACT FROM AUTHOR]

Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field F , the list needs to contain only finitely many F -representable matroids, due to the well-quasi-ordering of F -representable matroids of bounded branch-width under taking matroid minors [J.F. Geelen, A.M.H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these F -representable excluded minors in general. We consider the class of matroids of path-width at most k for fixed k. We prove that for a finite field F , every F -representable excluded minor for the class of matroids of path-width at most k has at most 2 | F | O (k 2) elements. We can therefore compute, for any integer k and a fixed finite field F , the set of F -representable excluded minors for the class of matroids of path-width k , and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an F -represented matroid is at most k. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most k has at most 2 2 O (k 2) vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs. [ABSTRACT FROM AUTHOR]

In this paper we used the cubic roots of unit group together with the concept of cordiality in graph theory to introduce a new method of labeling, this construed cubic cordial labeling can be applied to all paths, cycles, fans and wheel graphs. Moreover, some other properties are investigated and show that the union of any two cycles and the union any two paths are cubic cordial graphs. Also, we study the cubic cordiality for the union of any cycle with a path. [ABSTRACT FROM AUTHOR]

The tasks of increasing the degree of cohesion and convergence are among the key ones for the post-war recovery and development of regional economic systems. They aim of socio-economic cohesion increase is mentioned as the main in the State Strategy for Regional Development of Ukraine, in the Post-War Recovery Plan, at it brings national regional development goals closer to European ones. In order to implement the tasks of post-war recovery and further development of regional economic systems (RES) based on the effective use of available resources, it is necessary to develop a methodological toolkit, to define strategic principles for the construction and development of RES. The peculiarities of the operation of RES in the period of post-war recovery are indicated. The concepts of "solidarity from below" and "solidarity from above" are introduced. An analogy is drawn between the management of the RES and the coordination of the activities of enterprises within the framework of the CPFR strategy. The means of stimulating cooperation in the RES, in particular institutional mechanisms and information technologies, are considered. Simulation models of economic cooperative subsystems were built using the iThink program. [ABSTRACT FROM AUTHOR]

For any odd t ≥ 9 , we present a polynomial-time algorithm that solves the 3-colouring problem, and finds a 3-colouring if one exists, in P t -free graphs of odd girth at least t - 2 . In particular, our algorithm works for (P 9 , C 3 , C 5) -free graphs, thus making progress towards determining the complexity of 3-colouring in P t -free graphs, which is open for t ≥ 8 . [ABSTRACT FROM AUTHOR]

In this paper and a companion paper, we prove that, if m $m$ is sufficiently large, every graph on m+1 $m+1$ vertices that has a universal vertex and minimum degree at least ⌊2m3⌋ $\lfloor \phantom{\rule[-0.5em]{}{0ex}}\frac{2m}{3}\rfloor $ contains each tree T $T$ with m $m$ edges as a subgraph. Our result confirms, for large m $m$, an important special case of a recent conjecture by Havet, Reed, Stein and Wood. The present paper already contains an approximate version of the result. [ABSTRACT FROM AUTHOR]