29 results on '"Adjacency matrix"'
Search Results
2. Minimum ev-Dominating Energy of Semigraph.
- Author
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Nath, Niva Rani, Nath, Surajit Kumar, Nandi, Ardhendu Kumar, and Nath, Biswajit
- Subjects
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ABSOLUTE value , *LAPLACIAN matrices , *EIGENVALUES , *DOMINATING set - Abstract
This paper established the idea of minimum evdominating matrix of semigraph and calculated its energy. The minimum ev-dominating energy EmeD(G) of a semigraph G is the sum of the absolute values of the eigenvalues of the minimum ev-dominating matrix. Here some results are also derived in connection with the energy of minimum evdominating matrix. Some lower bounds are also established. [ABSTRACT FROM AUTHOR]
- Published
- 2024
3. ON THE SPREAD OF THE GENERALIZED ADJACENCY MATRIX OF A GRAPH.
- Author
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BAGHIPUR, M., GHORBANI, M., GANIE, H. A., and PIRZADA, S.
- Subjects
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GRAPH connectivity , *MATRICES (Mathematics) , *EIGENVALUES , *LAPLACIAN matrices - Abstract
Let D(G) and A(G), respectively, be the diagonal matrix of vertex de- grees and the adjacency matrix of a connected graph G. The generalized adjacency matrix of G is defined as Aα(G) = αD(G) + (1 − α)A(G), α ∈ [0, 1]. The spread of the generalized adjacency matrix, denoted by S(Aα(G)), is defined as the difference of the largest and smallest eigenvalues of Aα(G). The spread S(Q(G)) of Q(G), the signless Laplacian matrix of G and S(A(G)) of A(G) are similarly defined. In this paper, we establish the relationships between S(Aα(G)), S(Q(G)) and S(A(G)). Further, we find bounds for S(Aα(G)) involving different parameters of G. Also, we obtain lower bounds for S(Aα(G)) in terms of the clique number and independence number of G, and we characterize the extremal graphs in some cases. [ABSTRACT FROM AUTHOR]
- Published
- 2023
4. A Unified Framework for Optimization-Based Graph Coarsening.
- Author
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Kumar, Manoj, Sharma, Anurag, and Kumar, Sandeep
- Subjects
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MACHINE learning , *LAPLACIAN matrices , *MATHEMATICAL regularization - Abstract
Graph coarsening is a widely used dimensionality reduction technique for approaching large-scale graph machine-learning problems. Given a large graph, graph coarsening aims to learn a smaller-tractable graph while preserving the properties of the originally given graph. Graph data consist of node features and graph matrix (e.g., adjacency and Laplacian). The existing graph coarsening methods ignore the node features and rely solely on a graph matrix to simplify graphs. In this paper, we introduce a novel optimization-based framework for graph dimensionality reduction. The proposed framework lies in the unification of graph learning and dimensionality reduction. It takes both the graph matrix and the node features as the input and learns the coarsen graph matrix and the coarsen feature matrix jointly while ensuring desired properties. The proposed optimization formulation is a multi-block non-convex optimization problem, which is solved efficiently by leveraging block majorization-minimization, log determinant, Dirichlet energy, and regularization frameworks. The proposed algorithms are provably convergent and practically amenable to numerous tasks. It is also established that the learned coarsened graph is ϵ 2 (0, 1) similar to the original graph. Extensive experiments elucidate the efficacy of the proposed framework for real-world applications. The code for all the experimental results is available at CODE. [ABSTRACT FROM AUTHOR]
- Published
- 2023
5. SPECTRAL PROPERTIES OF THE NON{PERMUTABILITY GRAPH OF SUBGROUPS.
- Author
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MUHIE, SEID KASSAW
- Subjects
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FINITE groups , *SYLOW subgroups , *CHARTS, diagrams, etc. , *LAPLACIAN matrices , *REGULAR graphs - Abstract
Given a finite group G and the subgroups lattice L(G) of G, the \textit{non--permutability graph of subgroups} ΓL(G) is introduced as the graph with vertices in L(G)\CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and edges obtained by joining two vertices X,Y if XY≠YX. Here we study the behaviour of the non-permutability graph of subgroups using algebraic properties of associated matrices such as the adjacency and the Laplacian matrix. Further, we study the structure of some classes of groups whose non-permutability graph is strongly regular. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
6. Graph Theory: A Lost Component for Development in Nigeria.
- Author
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Babarinsa, Olayiwola
- Subjects
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GRAPH theory , *LAPLACIAN matrices , *MATHEMATICS , *RESEARCH - Abstract
Graph theory is one of the neglected branches of mathematics in Nigeria but with the most applications in other fields of research. This article shows the paucity, importance, and necessity of graph theory in the development of Nigeria. The adjacency matrix and dual graph of the Nigeria map were presented. The graph spectrum and energies (graph energy and Laplacian energy) of the dual graph were computed. Then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the Eulerian circuit and Hamiltonian paths from the dual graph were obtained and discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
7. A γ Eigenvalues of Zero Divisor Graph of Integer Modulo and Von Neumann Regular Rings.
- Author
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Rather, Bilal Ahmad, Ali, Fawad, Ullah, Asad, Fatima, Nahid, and Dad, Rahim
- Subjects
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EIGENVALUES , *DIVISOR theory , *RINGS of integers , *LAPLACIAN matrices , *SPECTRAL theory , *INTEGERS , *MATRICES (Mathematics) - Abstract
The A γ matrix of a graph G is determined by A γ (G) = (1 − γ) A (G) + γ D (G) , where 0 ≤ γ ≤ 1 , A (G) and D (G) are the adjacency and the diagonal matrices of node degrees, respectively. In this case, the A γ matrix brings together the spectral theories of the adjacency, the Laplacian, and the signless Laplacian matrices, and many more γ adjacency-type matrices. In this paper, we obtain the A γ eigenvalues of zero divisor graphs of the integer modulo rings and the von Neumann rings. These results generalize the earlier published spectral theories of the adjacency, the Laplacian and the signless Laplacian matrices of zero divisor graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
8. A Unified Framework for Structured Graph Learning via Spectral Constraints.
- Author
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Kumar, Sandeep, Jiaxi Ying, de M. Cardoso, Jose Vinícius, and Palomar, Daniel P.
- Subjects
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COMBINATORIAL optimization , *GRAPH theory , *REGULAR graphs , *SPECTRAL theory , *LAPLACIAN matrices , *NP-hard problems , *CONSTRAINT programming - Abstract
Graph learning from data is a canonical problem that has received substantial attention in the literature. Learning a structured graph is essential for interpretability and identification of the relationships among data. In general, learning a graph with a specific structure is an NP-hard combinatorial problem and thus designing a general tractable algorithm is challenging. Some useful structured graphs include connected, sparse, multi-component, bipartite, and regular graphs. In this paper, we introduce a unified framework for structured graph learning that combines Gaussian graphical model and spectral graph theory. We propose to convert combinatorial structural constraints into spectral constraints on graph matrices and develop an optimization framework based on block majorization-minimization to solve structured graph learning problem. The proposed algorithms are provably convergent and practically amenable for a number of graph based applications such as data clustering. Extensive numerical experiments with both synthetic and real data sets illustrate the effectiveness of the proposed algorithms. An open source R package containing the code for all the experiments is available at https://CRAN.R-project.org/package=spectralGraphTopology. [ABSTRACT FROM AUTHOR]
- Published
- 2020
9. Smith and critical groups of polar graphs.
- Author
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Pantangi, Venkata Raghu Tej and Sin, Peter
- Subjects
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LAPLACIAN matrices , *DIVISOR theory , *VECTOR spaces - Abstract
We compute the elementary divisors of the adjacency and Laplacian matrices of families of polar graphs. These graphs have as vertices the isotropic one-dimensional subspaces of finite vector spaces with respect to non-degenerate forms, with adjacency given by orthogonality. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
10. On the multiplicity of [formula omitted] as an [formula omitted]-eigenvalue of signed graphs with pendant vertices.
- Author
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Belardo, Francesco, Brunetti, Maurizio, and Ciampella, Adriana
- Subjects
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LAPLACIAN matrices , *MULTIPLICITY (Mathematics) , *GEOMETRIC vertices - Abstract
A signed graph is a pair Γ = (G , σ) , where x = (V (G) , E (G)) is a graph and σ : E (G) → { + 1 , − 1 } is the sign function on the edges of G. For any α ∈ [ 0 , 1 ] we consider the matrix A α (Γ) = α D (G) + (1 − α) A (Γ) , where D (G) is the diagonal matrix of the vertex degrees of G , and A (Γ) is the adjacency matrix of Γ. Let m A α (Γ) (α) be the multiplicity of α as an A α (Γ) -eigenvalue, and let G have p (G) pendant vertices, q (G) quasi-pendant vertices, and no components isomorphic to K 2. It is proved that m A α (Γ) (α) = p (G) − q (G) whenever all internal vertices are quasi-pendant. If this is not the case, it turns out that m A α (Γ) (α) = p (G) − q (G) + m N α (Γ) (α) , where m N (Γ) (α) denotes the multiplicity of α as an eigenvalue of the matrix N α (Γ) obtained from A α (Γ) taking the entries corresponding to the internal vertices which are not quasi-pendant. Such results allow to state a formula for the multiplicity of 1 as an eigenvalue of the Laplacian matrix L (Γ) = D (G) − A (Γ). Furthermore, it is detected a class G of signed graphs whose nullity – i.e. the multiplicity of 0 as an A (Γ) -eigenvalue – does not depend on the chosen signature. The class G contains, among others, all signed trees and all signed lollipop graphs. It also turns out that for signed graphs belonging to a subclass G ′ ⊂ G the multiplicity of 1 as Laplacian eigenvalue does not depend on the chosen signatures. Such subclass contains trees and circular caterpillars. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
11. The anti-adjacency matrix of a graph: Eccentricity matrix.
- Author
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Wang, Jianfeng, Lu, Mei, Belardo, Francesco, and Randić, Milan
- Subjects
- *
MOLECULAR graph theory , *EIGENVALUES , *LAPLACIAN matrices , *DISTANCE geometry , *TREE graphs - Abstract
Abstract In this paper we introduce a new graph matrix, named the anti-adjacency matrix or eccentricity matrix, which is constructed from the distance matrix of a graph by keeping for each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping for each row and each column only the distances equal to 1. We show that the eccentricity matrix of trees is irreducible, and we investigate the relations between the eigenvalues of the adjacency and eccentricity matrices. Finally, we give some applications of this new matrix in terms of molecular descriptors, and we conclude by proposing some further research problems. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
12. A characterization of oriented hypergraphic Laplacian and adjacency matrix coefficients.
- Author
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Chen, Gina, Liu, Vivian, Robinson, Ellen, Rusnak, Lucas J., and Wang, Kyle
- Subjects
- *
HYPERGRAPHS , *LAPLACIAN matrices , *COEFFICIENTS (Statistics) , *MATHEMATICAL bounds , *GENERALIZATION - Abstract
An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
13. On the α-index of graphs with pendent paths.
- Author
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Nikiforov, Vladimir and Rojo, Oscar
- Subjects
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GRAPH theory , *LINEAR algebra , *LAPLACIAN operator , *MATRICES (Mathematics) , *LAPLACIAN matrices - Abstract
Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For every real α ∈ [ 0 , 1 ] , write A α ( G ) for the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . This paper presents some extremal results about the spectral radius ρ α ( G ) of A α ( G ) that generalize previous results about ρ 0 ( G ) and ρ 1 / 2 ( G ) . In particular, write B p , q , r be the graph obtained from a complete graph K p by deleting an edge and attaching paths P q and P r to its ends. It is shown that if α ∈ [ 0 , 1 ) and G is a graph of order n and diameter at least k , then ρ α ( G ) ≤ ρ α ( B n − k + 2 , ⌊ k / 2 ⌋ , ⌈ k / 2 ⌉ ) , with equality holding if and only if G = B n − k + 2 , ⌊ k / 2 ⌋ , ⌈ k / 2 ⌉ . This result generalizes results of Hansen and Stevanović [5] , and Liu and Lu [7] . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
14. Signed graphs with cut points whose positive inertia indexes are two.
- Author
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Wang, Xinlei, Wong, Dein, and Tian, Fenglei
- Subjects
- *
LINEAR algebra , *GEOMETRIC vertices , *GRAPH theory , *LAPLACIAN matrices , *GRAPHIC methods - Abstract
A signed graph G σ consists of an underlying graph G and a sign function σ , which assigns each edge uv of G a sign σ ( u v ) , either positive or negative. The adjacency matrix of G σ is defined as A ( G σ ) = ( a u , v σ ) with a u , v σ = σ ( u v ) a u , v , where a u , v = 1 if u , v ∈ V ( G ) are adjacent, and a u , v = 0 otherwise. The positive inertia index of G σ , written as p ( G σ ) , is defined to be the number of positive eigenvalues of A ( G σ ) . Recently, Yu et al. (2016) [12] characterized the signed graphs G σ with pendant vertices such that p ( G σ ) = 2 . In this paper, we extend the above work to a more general case, characterizing the signed graphs G σ with cut points whose positive inertia index is 2. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
15. On the determinant of the Laplacian matrix of a complex unit gain graph.
- Author
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Wang, Yi, Gong, Shi-Cai, and Fan, Yi-Zheng
- Subjects
- *
GRAPH theory , *LAPLACIAN matrices , *MATHEMATICAL complexes , *UNDIRECTED graphs , *PATHS & cycles in graph theory - Abstract
Let G be a complex unit gain graph which is obtained from an undirected graph Γ by assigning a complex unit φ ( v i v j ) to each oriented edge v i v j such that φ ( v i v j ) φ ( v j v i ) = 1 for all edges. The Laplacian matrix of G is defined as L ( G ) = D ( G ) − A ( G ) , where D ( G ) is the degree diagonal matrix of Γ and A ( G ) = ( a i j ) has a i j = φ ( v i v j ) if v i is adjacent to v j and a i j = 0 otherwise. In this paper, we provide a combinatorial description of det ( L ( G ) ) that generalizes that for the determinant of the Laplacian matrix of a signed graph. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
16. On the critical group of the missing Moore graph.
- Author
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Ducey, Joshua E.
- Subjects
- *
EQUIVALENCE relations (Set theory) , *LAPLACIAN matrices , *JACOBIAN matrices , *REGULAR graphs , *MATHEMATICAL bounds - Abstract
We consider the critical group of a hypothetical Moore graph of diameter 2 and valency 57 . Determining this group is equivalent to finding the Smith normal form of the Laplacian matrix of such a graph. We show that all of the Sylow p -subgroups of the critical group must be elementary abelian with the exception of p = 5 . We prove that the 5 -rank of the Laplacian matrix determines the critical group up to two possibilities. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
17. A note on the positive semidefiniteness of Aα(G).
- Author
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Nikiforov, Vladimir and Rojo, Oscar
- Subjects
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SEMIDEFINITE programming , *GRAPH theory , *EIGENVALUES , *BIPARTITE graphs , *LAPLACIAN matrices - Abstract
Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For every real α ∈ [ 0 , 1 ] , write A α ( G ) for the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let α 0 ( G ) be the smallest α for which A α ( G ) is positive semidefinite. It is known that α 0 ( G ) ≤ 1 / 2 . The main results of this paper are: (1) if G is d -regular then α 0 = − λ min ( A ( G ) ) d − λ min ( A ( G ) ) , where λ min ( A ( G ) ) is the smallest eigenvalue of A ( G ) ; (2) G contains a bipartite component if and only if α 0 ( G ) = 1 / 2 ; (3) if G is r -colorable, then α 0 ( G ) ≥ 1 / r . [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
18. Relation between signless Laplacian energy, energy of graph and its line graph.
- Author
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Das, Kinkar Ch. and Mojallal, Seyed Ahmad
- Subjects
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LAPLACIAN matrices , *GRAPH theory , *ABSOLUTE value , *MATHEMATICAL bounds , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
The energy of a simple graph G , E ( G ) , is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of line graph and the signless Laplacian energy of graph G are denoted by E ( L G ) ( L G is the line graph of G ) and LE + ( G ) , respectively. In this paper we obtain a relation between E ( L G ) and LE + ( G ) of graph G . From this relation we characterize all the graphs satisfying E ( L G ) = LE + ( G ) + 4 m n − 4 . We also present a relation between E ( G ) and E ( L G ) . Moreover, we give an upper bound on E ( L G ) of graph G and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
19. Spectral characterizations of signed lollipop graphs.
- Author
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Belardo, Francesco and Petecki, Paweł
- Subjects
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GRAPH theory , *LAPLACIAN matrices , *MATHEMATICAL analysis , *BOUNDARY value problems , *NUMERICAL analysis - Abstract
Let Γ = ( G , σ ) be a signed graph, where G is the underlying simple graph and σ : E ( G ) → { + , − } is the sign function on the edges of G . In this paper we consider the spectral characterization problem extended to the adjacency matrix and Laplacian matrix of signed graphs. After giving some basic results, we study the spectral determination of signed lollipop graphs, and we show that any signed lollipop graph is determined by the spectrum of its Laplacian matrix. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
20. The largest Laplacian and adjacency indices of complete caterpillars of fixed diameter.
- Author
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Abreu, Nair, Rojo, Oscar, and Lenes, Eber
- Subjects
- *
LAPLACIAN matrices , *CATERPILLARS , *GEOMETRIC vertices , *MATHEMATICS , *INTEGERS - Abstract
A complete caterpillar is a caterpillar in which each internal vertex is a quasi-pendent vertex. In this paper, in the class of all complete caterpillars on n vertices and diameter d, the caterpillar attaining the largest Laplacian index is determined. In addition, it is proved that this caterpillar also attains the largest adjacency index. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
21. Notes on the second largest eigenvalue of a graph.
- Author
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Simić, Slobodan K., Anđelić, Milica, da Fonseca, Carlos M., and Živković, Dejan
- Subjects
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EIGENVALUES , *GRAPH theory , *LAPLACIAN matrices , *VERTEX operator algebras , *POLYNOMIALS - Abstract
For a fixed real number r we give several necessary and/or sufficient conditions for a graph to have the second largest eigenvalue of the adjacency matrix, or signless Laplacian matrix, less then or equal to r . [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
22. Sharp bounds for the spectral radius of nonnegative matrices.
- Author
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Xing, Rundan and Zhou, Bo
- Subjects
- *
MATHEMATICAL bounds , *RADIUS (Geometry) , *NONNEGATIVE matrices , *GENERALIZABILITY theory , *LAPLACIAN matrices - Abstract
Abstract: We give sharp upper and lower bounds for the spectral radius of a nonnegative matrix with all row sums positive using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We compare these bounds with the known bounds using the row sums by examples. We also apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix and some distance-based matrices. Some known results are generalized and improved. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
23. On two conjectures of Randić index and the largest signless Laplacian eigenvalue of graphs.
- Author
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Deng, Hanyuan, Balachandran, S., and Ayyaswamy, S.K.
- Subjects
- *
EIGENVALUES , *GRAPH theory , *LAPLACIAN matrices , *HARMONIC analysis (Mathematics) , *MATHEMATICAL proofs , *MATHEMATICAL analysis - Abstract
Abstract: The Randić index R of a graph G is defined as the sum of over all edges of G, where denotes the degree of a vertex in G. is the largest eigenvalue of the signless Laplacian matrix of G, where D is the diagonal matrix with degrees of the vertices on the main diagonal and A is the adjacency matrix of G. Hansen and Lucas [18] conjectured (1) and equality holds for and (2) with equality if and only if for and for , respectively. In this paper, we prove the conjecture (1) and obtain a result very close to the conjecture (2). Moreover, we give some results relating harmonic index and the largest eigenvalue of the adjacency matrix. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
24. Sharp bounds on the spectral radius of a nonnegative matrix.
- Author
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Duan, Xing and Zhou, Bo
- Subjects
- *
MATHEMATICAL bounds , *RADIUS (Geometry) , *MATRICES (Mathematics) , *GRAPH theory , *LAPLACIAN matrices , *GENERALIZATION - Abstract
Abstract: We give upper and lower bounds for the spectral radius of a nonnegative matrix using its row sums and characterize the equality cases if the matrix is irreducible. Then we apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix. Some known results in the literature are generalized and improved. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
25. Nonpositive eigenvalues of the adjacency matrix and lower bounds for Laplacian eigenvalues.
- Author
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Charles, Zachary B., Farber, Miriam, Johnson, Charles R., and Kennedy-Shaffer, Lee
- Subjects
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EIGENVALUES , *MATRICES (Mathematics) , *MATHEMATICAL bounds , *LAPLACIAN matrices , *UNDIRECTED graphs , *GRAPH theory - Abstract
Abstract: Let be the smallest number such that the adjacency matrix of any undirected graph with vertices or more has at least nonpositive eigenvalues. We show that is well-defined and prove that the values of for are 1, 3, 6, 10, 16 respectively. In addition, we prove that for all , , in which is the Ramsey number for and , and is the triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the largest eigenvalue is bounded from below the largest degree, which generalizes some prior results. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
26. An upper bound on the largest signless Laplacian of an odd unicyclic graph.
- Author
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COLLAO, MACARENA, PIZARRO, PAMELA, and ROJO, OSCAR
- Subjects
- *
MATHEMATICAL bounds , *LAPLACIAN matrices , *GRAPH theory , *EIGENVALUES , *TREE graphs , *UNIQUENESS (Mathematics) , *GENERALIZATION - Abstract
We derive an upper bound on the largest signless Laplacian eigenvalue of an odd unicyclic graph. The bound is given in terms of the largest vertex degree and the largest height of the trees obtained removing the edges of the unique cycle in the graph. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
27. A Note on the Estrada Index of the A α -Matrix.
- Author
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Rodríguez, Jonnathan, Nina, Hans, and Sztrik, János
- Subjects
- *
LAPLACIAN matrices , *EIGENVALUES - Abstract
Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A (G) and D (G) and defined the A α -matrix for every real α ∈ [ 0 , 1 ] as: A α (G) = α D (G) + (1 − α) A (G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the A α -matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
28. On the Aα-Spectral Radii of Cactus Graphs.
- Author
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Wang, Chunxiang, Wang, Shaohui, Liu, Jia-Bao, and Wei, Bing
- Subjects
- *
LAPLACIAN matrices , *GRAPH connectivity , *CACTUS , *EIGENVALUES - Abstract
Let A (G) be the adjacent matrix and D (G) the diagonal matrix of the degrees of a graph G, respectively. For 0 ≤ α ≤ 1 , the A α -matrix is the general adjacency and signless Laplacian spectral matrix having the form of A α (G) = α D (G) + (1 − α) A (G) . Clearly, A 0 (G) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A α -spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
29. The Characteristic Polynomials of Symmetric Graphs.
- Author
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Chbili, Nafaa, Al Dhaheri, Shamma, Tahnon, Mei Y., and Abunamous, Amna A. E.
- Subjects
- *
POLYNOMIALS , *MATHEMATICAL symmetry , *GRAPH theory , *INVARIANTS (Mathematics) , *AUTOMORPHISMS , *FINITE fields , *CONGRUENCE lattices , *LAPLACIAN matrices - Abstract
In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ , with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ [ F ] by one of Γ \ F . A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
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