43 results on '"Mathematics - Combinatorics"'
Search Results
2. Expansion in matrix-weighted graphs
- Author
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Jakob Hansen
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Positive-definite matrix ,Edge (geometry) ,Graph ,Mathematics - Spectral Theory ,Combinatorics ,Matrix (mathematics) ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Expander graph ,Combinatorics (math.CO) ,Geometry and Topology ,Adjacency matrix ,Spectral Theory (math.SP) ,Laplace operator ,Expander mixing lemma ,Mathematics - Abstract
A matrix-weighted graph is an undirected graph with a $k\times k$ positive semidefinite matrix assigned to each edge. There are natural generalizations of the Laplacian and adjacency matrices for such graphs. These matrices can be used to define and control expansion for matrix-weighted graphs. In particular, an analogue of the expander mixing lemma and one half of a Cheeger-type inequality hold for matrix-weighted graphs. A new definition of a matrix-weighted expander graph suggests the tantalizing possibility of families of matrix-weighted graphs with better-than-Ramanujan expansion., 21 pages, 2 figures
- Published
- 2021
3. Spectral radius and clique partitions of graphs
- Author
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Edwin van Dam, Jiang Zhou, Econometrics and Operations Research, and Research Group: Operations Research
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Degree (graph theory) ,Spectral radius ,Block (permutation group theory) ,Clique partition ,Clique (graph theory) ,Upper and lower bounds ,Graph ,Combinatorics ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Steiner 2-design ,FOS: Mathematics ,Clique partition number ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,05C50, 05C70, 05B20 ,Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint t-cliques. The extremal graphs attaining the bounds are exactly the block graphs of Steiner 2-designs and the regular graphs with K t -decompositions, respectively.
- Published
- 2021
4. Hoffman's ratio bound
- Author
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Willem H. Haemers and Econometrics and Operations Research
- Subjects
Clique ,Eigenvalue ,010103 numerical & computational mathematics ,Clique (graph theory) ,01 natural sciences ,Upper and lower bounds ,Graph ,Hoffman bound ,Combinatorics ,Mathematics::Probability ,Computer Science::Discrete Mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Adjacency matrix ,0101 mathematics ,Eigenvalues and eigenvectors ,Independence number ,Mathematics ,Numerical Analysis ,Mathematics::Combinatorics ,Algebra and Number Theory ,010102 general mathematics ,Coclique ,Black hole ,Graph (abstract data type) ,Regular graph ,Combinatorics (math.CO) ,Mathematics::Differential Geometry ,Geometry and Topology - Abstract
Hoffman's ratio bound is an upper bound for the independence number of a regular graph in terms of the eigenvalues of the adjacency matrix. The bound has proved to be very useful and has been applied many times. Hoffman did not publish his result, and for a great number of users the emergence of Hoffman's bound is a black hole. With his note I hope to clarify the history of this bound and some of its generalizations.
- Published
- 2021
5. Integer Laplacian eigenvalues of chordal graphs
- Author
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Lilian Markenzon, Claudia Marcela Justel, and Nair Maria Maia de Abreu
- Subjects
FOS: Computer and information sciences ,Numerical Analysis ,Algebra and Number Theory ,Discrete Mathematics (cs.DM) ,010102 general mathematics ,010103 numerical & computational mathematics ,Mathematics::Spectral Theory ,01 natural sciences ,Laplacian eigenvalues ,Graph ,Vertex (geometry) ,Combinatorics ,Computer Science::Discrete Mathematics ,Chordal graph ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Algebraic number ,Laplace operator ,Eigenvalues and eigenvectors ,Computer Science - Discrete Mathematics ,Mathematics - Abstract
In this paper, structural properties of chordal graphs are analysed, in order to establish a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we stablish a sufficient condition for the cardinality of a maximal clique to appear as an integer Laplacian eigenvalue. Finally, we review two subclasses of chordal graphs, showing for them some new properties., 15 pages, 5 figures
- Published
- 2021
6. Edge-matching graph contractions and their interlacing properties
- Author
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Noam Leiter and Daniel Zelazo
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Mathematics::Classical Analysis and ODEs ,Interlacing ,Mathematics::Spectral Theory ,Minimax ,Graph ,Mathematics - Spectral Theory ,Combinatorics ,Mathematics::Probability ,FOS: Mathematics ,Graph reduction ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Symmetric matrix ,Combinatorics (math.CO) ,Geometry and Topology ,Spectral Theory (math.SP) ,Laplace operator ,Subspace topology ,Eigenvalues and eigenvectors ,Mathematics - Abstract
For a given graph $\mathcal{G}$ of order $n$ with $m$ edges, and a real symmetric matrix associated to the graph, $M\left(\mathcal{G}\right)\in\mathbb{R}^{n\times n}$, the interlacing graph reduction problem is to find a graph $\mathcal{G}_{r}$ of order $r
- Published
- 2021
7. Connectivity for quantum graphs
- Author
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Andrew T. Swift and Javier Alejandro Chávez-Domínguez
- Subjects
Social connectedness ,010103 numerical & computational mathematics ,Information theory ,01 natural sciences ,05C40 (Primary) 47L25, 81P45 (Secondary) ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Quantum information ,Operator Algebras (math.OA) ,Quantum ,Mathematics ,Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,Mathematics - Operator Algebras ,Graph theory ,Graph ,ComputerSystemsOrganization_MISCELLANEOUS ,Quantum graph ,Combinatorics (math.CO) ,Geometry and Topology ,Hamming code ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In quantum information theory there is a construction for quantum channels, appropriately called a quantum graph, that generalizes the confusability graph construction for classical channels in classical information theory. In this paper, we provide a definition of connectedness for quantum graphs that generalizes the classical definition. This is used to prove a quantum version of a particular case of the classical tree-packing theorem from graph theory. Generalizations for the related notions of $k$-connectedness and of orthogonal representation are also proposed for quantum graphs, and it is shown that orthogonal representations have the same implications for connectedness as they do in the classical case., Comment: 15 pages
- Published
- 2021
8. Integral graphs obtained by dual Seidel switching
- Author
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Da Zhao, Elena V. Konstantinova, Sergey Goryainov, and Hong-Hai Li
- Subjects
Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,05C25, 05E10, 05E15 ,Graph ,FOS: Mathematics ,Bipartite graph ,Odd graph ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Invariant (mathematics) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Dual Seidel switching is a graph operation introduced by W. Haemers in 1984. This operation can change the graph, however it does not change its bipartite double, and because of this, the operation leaves the squares of the eigenvalues invariant. Thus, if a graph is integral then it is still integral after dual Seidel switching. In this paper two new infinite families of integral graphs are obtained by applying dual Seidel switching to the Star graphs and the Odd graphs. In particular, three new 4-regular integral graphs with their spectra are found.
- Published
- 2020
9. Perfect state transfer on oriented graphs
- Author
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Chris Godsil and Sabrina Lato
- Subjects
Discrete mathematics ,Quantum Physics ,Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,Graph ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Adjacency list ,Symmetric matrix ,Quantum walk ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Perfect state transfer ,Laplacian matrix ,Quantum Physics (quant-ph) ,Undirected graph ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
Quantum walks on undirected graphs have been studied using symmetric matrices, such as the adjacency or Laplacian matrix, and many results about perfect state transfer are known. We extend some of those results to oriented graphs. We also study the phenomena, unique to oriented graphs, of multiple state transfer, where there is a set of vertices such that perfect state transfer occurs between every pair in that set. We give a characterization of multiple state transfer, and a new example of a graph where it occurs., Comment: 17 pages, 3 figures
- Published
- 2020
10. The asymptotic value of energy for matrices with degree-distance-based entries of random graphs
- Author
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Zhiqian Wang, Yiyang Li, and Xueliang Li
- Subjects
Weighted distance ,Random graph ,Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Graph ,05C50, 05C80, 05C22, 92E10 ,Symmetric function ,Combinatorics ,Matrix (mathematics) ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Distance based ,Mathematics - Abstract
For a graph $G=(V, E)$ and $i, j\in V$, denote the distance between $i$ and $j$ in $G$ by $D(i, j)$ and the degrees of $i$, $j$ by $d_i$, $d_j$, respectively. Let $f(D(i, j), d_{i}, d_{j})$ be a function symmetric in $i$ and $j$. Define a matrix $W_f(G)$, called the weighted distance matrix, of $G$, with the $ij$-entry $W_f(G)(i, j)=f(D(i, j), d_{i}, d_{j})$ if $i\neq j$ and $W_f(G)(i, j)=0$ if $i=j$. In this paper, we prove that if the symmetric function $f$ satisfies that $f(D(i, j), (1+o(1))np, (1+o(1))np)=(1+o(1))f(D(i, j), np, np)$, then for almost all graphs $G_p$ in the $Erd\ddot{o}s$-$R\acute{e}nyi$ random graph model $\mathcal{G}_{n, p}$, the energy of $W_f(G_p)$ is $\{(\frac{8}{3\pi}\sqrt{p(1-p)}+o(1))\cdot|f(1, np, np)-f(2, np, np)|+o(|f(2, np, np)|)\}\cdot n^{3/2}$. As a consequence, we give the asymptotic values of energies of a variety of weighted distance matrices with function $f$ from distance-based only and mixed with degree-distance-based topological indices of chemical use. This generalizes our former result with only degree-based weights., Comment: 15 pages
- Published
- 2020
11. On graphs with 2 trivial distance ideals
- Author
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Carlos A. Alfaro
- Subjects
Numerical Analysis ,Algebra and Number Theory ,05C25, 05C50, 05E99, 13P15, 15A03, 68W30 ,010102 general mathematics ,010103 numerical & computational mathematics ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Graph ,Combinatorics ,Distance matrix ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Smith normal form - Abstract
Distance ideals generalize the Smith normal form of the distance matrix of a graph. The family of graphs with 2 trivial distance ideals contains the family of graphs whose distance matrix has at most 2 invariant factors equal to 1. Here we give an infinite family of forbidden induced subgraphs for the graphs with 2 trivial distance ideals. These are also related with other well known graph classes., Some typos were corrected
- Published
- 2020
12. On the graph Laplacian and the rankability of data
- Author
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Thomas R. Cameron, Amy N. Langville, and Heather C. Smith
- Subjects
Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,Laplacian spectrum ,010102 general mathematics ,010103 numerical & computational mathematics ,Directed graph ,01 natural sciences ,Graph ,Mathematics - Spectral Theory ,Hausdorff distance ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,90C35, 05C20, 05C22, 05C50, 62F07, 47A55 ,Tournament ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Laplacian matrix ,Spectral Theory (math.SP) ,MathematicsofComputing_DISCRETEMATHEMATICS ,College football ,Mathematics - Abstract
Recently, Anderson et al. (2019) proposed the concept of rankability, which refers to a dataset's inherent ability to produce a meaningful ranking of its items. In the same paper, they proposed a rankability measure that is based on an integer program for computing the minimum number of edge changes made to a directed graph in order to obtain a complete dominance graph, i.e., an acyclic tournament graph. In this article, we prove a spectral-degree characterization of complete dominance graphs and apply this characterization to produce a new measure of rankability that is cost-effective and more widely applicable. We support the details of our algorithm with several results regarding the conditioning of the Laplacian spectrum of complete dominance graphs and the Hausdorff distance between their Laplacian spectrum and that of an arbitrary directed graph with weights between zero and one. Finally, we analyze the rankability of datasets from the world of chess and college football.
- Published
- 2020
13. The role of the anti-regular graph in the spectral analysis of threshold graphs
- Author
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Matthew Ficarra, Brittany Sullivan, Cesar O. Aguilar, and Natalie Schurman
- Subjects
Numerical Analysis ,Threshold graph ,Algebra and Number Theory ,Conjecture ,010102 general mathematics ,Spectral properties ,Interlacing ,010103 numerical & computational mathematics ,01 natural sciences ,Graph ,Combinatorics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Regular graph ,Spectral analysis ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The purpose of this paper is to highlight the role played by the anti-regular graph within the class of threshold graphs. Using the fact that every threshold graph contains a maximal anti-regular graph, we show that some known results, and new ones, on the spectral properties of threshold graphs can be deduced from (i) the known results on the eigenvalues of anti-regular graphs, (ii) the subgraph structure of threshold graphs, and (iii) eigenvalue interlacing. In particular, we prove a strengthened version of the recently proved fact that no threshold graph contains an eigenvalue in the interval Ω = [ − 1 − 2 2 , − 1 + 2 2 ] , except possibly the trivial eigenvalues −1 and/or 0, determine the inertia of a threshold graph, and give partial results on a conjecture regarding the optimality of the non-trivial eigenvalues of an anti-regular graph within the class of threshold graphs.
- Published
- 2020
14. Grassmann graphs, degenerate DAHA, and non-symmetric dual q-Hahn polynomials
- Author
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Jae-Ho Lee
- Subjects
Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,Non symmetric ,Degenerate energy levels ,010103 numerical & computational mathematics ,01 natural sciences ,Linear subspace ,Graph ,Vertex (geometry) ,Combinatorics ,Finite field ,Hahn polynomials ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,05E30, 20C08, 33D45, 33D80 ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Vector space ,Mathematics - Abstract
We discuss the Grassmann graph $J_q(N,D)$ with $N \geq 2D$, having as vertices the $D$-dimensional subspaces of an $N$-dimensional vector space over the finite field $\mathbb{F}_q$. This graph is distance-regular with diameter $D$; to avoid trivialities we assume $D\geq 3$. Fix a pair of a Delsarte clique $C$ of $J_q(N,D)$ and a vertex $x$ in $C$. We construct a $2D$-dimensional irreducible module $\mathbf{W}$ for the Terwilliger algebra $\mathbf{T}$ of $J_q(N,D)$ associated with the pair $x$, $C$. We show that $\mathbf{W}$ is an irreducible module for the confluent Cherednik algebra $\mathcal{H}_\mathrm{V}$ and describe how the $\mathbf{T}$-action on $\mathbf{W}$ is related to the $\mathcal{H}_\mathrm{V}$-action on $\mathbf{W}$. Using the $\mathcal{H}_\mathrm{V}$-module $\mathbf{W}$, we define non-symmetric dual $q$-Hahn polynomials and prove their recurrence and orthogonality relations from a combinatorial viewpoint., Comment: 31 pages, 3 figures
- Published
- 2020
15. The energy of random signed graph
- Author
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Shuchao Li and Shujing Wang
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Positive edge ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Graph ,Combinatorics ,Multipartite ,05C50, 15A48 ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Adjacency matrix ,0101 mathematics ,Signed graph ,Eigenvalues and eigenvectors ,Mathematics - Abstract
A signed graph $\Gamma(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $\Gamma(G)$. The energy of a signed graph $\Gamma(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(\Gamma(G))$ of $\Gamma(G)$. The random signed graph model $\mathcal{G}_n(p, q)$ is defined as follows: Let $p, q \ge 0$ be fixed, $0 \le p+q \le 1$. Given a set of $n$ vertices, between each pair of distinct vertices there is either a positive edge with probability $p$ or a negative edge with probability $q$, or else there is no edge with probability $1-(p+ q)$. The edges between different pairs of vertices are chosen independently. In this paper, we obtain an exact estimate of energy for almost all signed graphs. Furthermore, we establish lower and upper bounds to the energy of random multipartite signed graphs., Comment: 11 page, 0 figures. arXiv admin note: text overlap with arXiv:0909.4923 by other authors
- Published
- 2020
16. Coulson integral formula for the vertex energy of a graph
- Author
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Marcelino Ramírez Ibáñez, Beatriz Carely Luna-Olivera, and Octavio Arizmendi
- Subjects
Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,Electron ,Hückel method ,01 natural sciences ,Graph ,Vertex (geometry) ,Mathematics - Spectral Theory ,FOS: Mathematics ,Theoretical chemistry ,Bipartite graph ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Molecular orbital ,Combinatorics (math.CO) ,Geometry and Topology ,Integral formula ,0101 mathematics ,Spectral Theory (math.SP) ,Computer Science::Databases ,Mathematics - Abstract
In this note we prove that the vertex energy of a graph, as defined in Arizmendi and Juarez (2018), can be calculated in terms of a Coulson integral formula. We present examples of how this formula can be used, and we show some applications to bipartite graphs., 14 pages, 5 figures
- Published
- 2019
17. Complete multipartite graphs that are determined, up to switching, by their Seidel spectrum
- Author
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Ranveer Singh, Abraham Berman, Xiao-Dong Zhang, and Naomi Shaked-Monderer
- Subjects
Vertex (graph theory) ,Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,Distance spectrum ,010103 numerical & computational mathematics ,Computer Science::Numerical Analysis ,01 natural sciences ,Graph ,Combinatorics ,Multipartite ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Partition (number theory) ,Adjacency list ,Combinatorics (math.CO) ,05C50, 05C12 ,Geometry and Topology ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained from $G$ by Seidel switching has the same Seidel spectrum. We consider $G$ to be determined by its Seidel spectrum, up to switching, if any graph with the same spectrum is switching equivalent to a graph isomorphic to $G$. It is shown that any graph which has the same spectrum as a complete $k$-partite graph is switching equivalent to a complete $k$-partite graph, and if the different partition sets sizes are $p_1,\ldots, p_l$, and there are at least three partition sets of each size $p_i$, $i=1,\ldots, l$, then $G$ is determined, up to switching, by its Seidel spectrum. Sufficient conditions for a complete tripartite graph to be determined by its Seidel spectrum are discussed, and a conjecture is made on complete tripartite graphs on more than 18 vertices., 12 page 2 figures
- Published
- 2019
18. Eigenvalue location in graphs of small clique-width
- Author
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Vilmar Trevisan, Martin Fürer, Carlos Hoppen, and David P. Jacobs
- Subjects
Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Graph ,Combinatorics ,Diagonal matrix ,Clique-width ,FOS: Mathematics ,Mathematics - Combinatorics ,05C50, 05C85, 15A18 ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Adjacency matrix ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Finding a diagonal matrix congruent to $A - cI$ for constants $c$, where $A$ is the adjacency matrix of a graph $G$ allows us to quickly tell the number of eigenvalues in a given interval. If $G$ has clique-width $k$ and a corresponding $k$-expression is known, then diagonalization can be done in time $O(\text{poly}(k) n)$ where $n$ is the order of $G$., Comment: 25 pages
- Published
- 2019
19. Spectral characterizations of anti-regular graphs
- Author
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Joon-yeob Lee, Eric Piato, Barbara J. Schweitzer, and Cesar O. Aguilar
- Subjects
Numerical Analysis ,Threshold graph ,Chebyshev polynomials ,Algebra and Number Theory ,0211 other engineering and technologies ,Asymptotic distribution ,021107 urban & regional planning ,010103 numerical & computational mathematics ,02 engineering and technology ,Mathematics::Spectral Theory ,01 natural sciences ,Graph ,Combinatorics ,05C50, 15B05, 05C75, 15A18 ,FOS: Mathematics ,Bipartite graph ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Trigonometry ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We study the eigenvalues of the unique connected anti-regular graph $A_n$. Using Chebyshev polynomials of the second kind, we obtain a trigonometric equation whose roots are the eigenvalues and perform elementary analysis to obtain an almost complete characterization of the eigenvalues. In particular, we show that the interval $\Omega=[\tfrac{-1-\sqrt{2}}{2}, \tfrac{-1+\sqrt{2}}{2}]$ contains only the trivial eigenvalues $\lambda = -1$ or $\lambda=0$, and any closed interval strictly larger than $\Omega$ will contain eigenvalues of $A_n$ for all $n$ sufficiently large. We also obtain bounds for the maximum and minimum eigenvalues, and for all other eigenvalues we obtain interval bounds that improve as $n$ increases. Moreover, our approach reveals a more complete picture of the bipartite character of the eigenvalues of $A_n$, namely, as $n$ increases the eigenvalues are (approximately) symmetric about the number $-\tfrac{1}{2}$. We also obtain an asymptotic distribution of the eigenvalues as $n\rightarrow\infty$. Finally, the relationship between the eigenvalues of $A_n$ and the eigenvalues of a general threshold graph is discussed., Comment: 20 pages, 6 figures, 1 table
- Published
- 2018
20. A characterization of oriented hypergraphic Laplacian and adjacency matrix coefficients
- Author
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Ellen Robinson, Kyle Wang, Gina Chen, Lucas J. Rusnak, and Vivian Liu
- Subjects
Hypergraph ,0211 other engineering and technologies ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,Incidence structure ,Computer Science::Discrete Mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Adjacency matrix ,05C50, 05C65, 05C22 ,Mathematics ,Discrete mathematics ,Numerical Analysis ,Mathematics::Combinatorics ,Algebra and Number Theory ,021107 urban & regional planning ,Graph ,010201 computation theory & mathematics ,Substructure ,Combinatorics (math.CO) ,Geometry and Topology ,Laplacian matrix ,Laplace operator - 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.
- Published
- 2018
21. The algebraic connectivity of a graph and its complement
- Author
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B. Afshari, Saieed Akbari, Bojan Mohar, and M.J. Moghaddamzadeh
- Subjects
Numerical Analysis ,Algebraic connectivity ,Algebra and Number Theory ,010103 numerical & computational mathematics ,0102 computer and information sciences ,16. Peace & justice ,01 natural sciences ,Graph ,Combinatorics ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,05C50 ,0101 mathematics ,Laplace operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
For a graph G, let λ 2 ( G ) denote its second smallest Laplacian eigenvalue. It was conjectured that λ 2 ( G ) + λ 2 ( G ‾ ) ≥ 1 , where G ‾ is the complement of G. In this paper, it is shown that max { λ 2 ( G ) , λ 2 ( G ‾ ) } ≥ 2 5 .
- Published
- 2018
22. The critical group of the Kneser graph on 2-subsets of an n-element set
- Author
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Ian Hill, Joshua E. Ducey, and Peter Sin
- Subjects
Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Graph ,Combinatorics ,Critical group ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Kneser graph ,Combinatorics (math.CO) ,Geometry and Topology ,05C50 ,0101 mathematics ,Laplacian matrix ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics ,Smith normal form - Abstract
In this paper we compute the critical group of the Kneser graph $KG(n,2)$. This is equivalent to computing the Smith normal form of a Laplacian matrix of this graph., The statement of Lemma 2.2, part (4) was missing a case. This has been corrected, and it has no effect on the rest of the paper
- Published
- 2018
23. Arithmetical structures on graphs
- Author
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Carlos E. Valencia and Hugo Corrales
- Subjects
11D72, 15B48 (Primary), 05C50, 05C76, 11D45, 11B83, 05E99, 11D68, 11C20 (Secondary) ,Vertex (graph theory) ,0102 computer and information sciences ,Algebraic geometry ,01 natural sciences ,Catalan number ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Arithmetic function ,Number Theory (math.NT) ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,0101 mathematics ,Subdivision ,Mathematics ,Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,Mathematics - Number Theory ,business.industry ,010102 general mathematics ,Graph ,010201 computation theory & mathematics ,Combinatorics (math.CO) ,Geometry and Topology ,business ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
Arithmetical structures on a graph were introduced by Lorenzini as some intersection matrices that arise in the study of degenerating curves in algebraic geometry. In this article we study these arithmetical structures, in particular we are interested in the arithmetical structures on complete graphs, paths, and cycles. We begin by looking at the arithmetical structures on a multidigraph from the general perspective of $M$-matrices. As an application, we recover the result of Lorenzini about the finiteness of the number of arithmetical structures on a graph. We give a description on the arithmetical structures on the graph obtained by merging and splitting a vertex of a graph in terms of its arithmetical structures. On the other hand, we give a description of the arithmetical structures on the clique--star transform of a graph, which generalizes the subdivision of a graph. As an application of this result we obtain an explicit description of all the arithmetical structures on the paths and cycles and we show that the number of the arithmetical structures on a path is a Catalan number., 23 pages, 5 Figures, Minor changes
- Published
- 2018
24. Laplacian spectral characterization of roses
- Author
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Changxiang He, Edwin van Dam, Research Group: Operations Research, and Econometrics and Operations Research
- Subjects
Vertex (graph theory) ,Closed walks ,Matchings ,010103 numerical & computational mathematics ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,Laplacian spectrum ,Sachs' theorem ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Adjacency matrix ,0101 mathematics ,Mathematics ,Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,Conjecture ,Graph ,Rose graphs ,010201 computation theory & mathematics ,Geometry and Topology ,Combinatorics (math.CO) ,Laplacian matrix ,05C50 ,Laplace operator - Abstract
A rose graph is a graph consisting of cycles that all meet in one vertex. We show that except for two specific examples, these rose graphs are determined by the Laplacian spectrum, thus proving a conjecture posed by Lui and Huang [F.J. Liu and Q.X. Huang, Laplacian spectral characterization of 3-rose graphs, Linear Algebra Appl. 439 (2013), 2914--2920]. We also show that if two rose graphs have a so-called universal Laplacian matrix with the same spectrum, then they must be isomorphic. In memory of Horst Sachs (1927-2016), we show the specific case of the latter result for the adjacency matrix by using Sachs' theorem and a new result on the number of matchings in the disjoint union of paths.
- Published
- 2018
25. A construction of distance cospectral graphs
- Author
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Kristin Heysse
- Subjects
Numerical Analysis ,Algebra and Number Theory ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,0102 computer and information sciences ,Mathematical proof ,01 natural sciences ,Graph ,Combinatorics ,Distance matrix ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Symmetric matrix ,Pairwise comparison ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Row ,Connectivity ,Eigenvalues and eigenvectors ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in their edge counts yet are cospectral with respect to the distance matrix. Further, we identify a subgraph switching behavior which constructs additional distance cospectral graphs. The proofs for both constructions rely on a perturbation of (most of) the distance eigenvectors of one graph to yield the distance eigenvectors of the other., Comment: 17 pages, 4 figures
- Published
- 2017
26. Bounds for the positive and negative inertia index of a graph
- Author
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Yi-Zheng Fan and Long Wang
- Subjects
Numerical Analysis ,Algebra and Number Theory ,0211 other engineering and technologies ,021107 urban & regional planning ,Circuit rank ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Graph ,Combinatorics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Adjacency matrix ,05C50 ,0101 mathematics ,Mathematics - Abstract
Let G be a graph and let A ( G ) be adjacency matrix of G. The positive inertia index (respectively, the negative inertia index) of G, denoted by p ( G ) (respectively, n ( G ) ), is defined to be the number of positive eigenvalues (respectively, negative eigenvalues) of A ( G ) . In this paper, we present the bounds for p ( G ) and n ( G ) as follows: m ( G ) − c ( G ) ≤ p ( G ) ≤ m ( G ) + c ( G ) , m ( G ) − c ( G ) ≤ n ( G ) ≤ m ( G ) + c ( G ) , where m ( G ) and c ( G ) are respectively the matching number and the cyclomatic number of G. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds respectively.
- Published
- 2017
27. Hypergraphs and hypermatrices with symmetric spectrum
- Author
-
Vladimir Nikiforov
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Conjecture ,010103 numerical & computational mathematics ,0102 computer and information sciences ,01 natural sciences ,Graph ,Combinatorics ,Matrix (mathematics) ,Monotone polygon ,010201 computation theory & mathematics ,FOS: Mathematics ,Bipartite graph ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Adjacency matrix ,0101 mathematics ,Eigenvalues and eigenvectors ,05C65 ,Mathematics - Abstract
It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to hypermatrices. To this end the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is shown that a nonnegative symmetric $r$-matrix $A$ has a symmetric spectrum if and only if $r$ is even and $A$ is odd-colorable. This result also solves a problem of Pearson and Zhang about hypergraphs with symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu. Separately, similar results are obtained for the $H$-spectram of hypermatrices., 17 pages. Corrected proof on p. 13
- Published
- 2017
28. On the distance spectra of graphs
- Author
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Leslie Hogben, Michael Tait, Jephian C.-H. Lin, Jessica De Silva, Aida Abiad, Kristin Heysse, Ghodratollah Aalipour, Jay Cummings, Franklin H. J. Kenter, Zhanar Berikkyzy, Wei Gao, QE Operations research, and RS: GSBE ETBC
- Subjects
Inertia ,Eigenvalue ,Determinant ,010103 numerical & computational mathematics ,0102 computer and information sciences ,01 natural sciences ,Graph ,Combinatorics ,Indifference graph ,Double odd graph ,Chordal graph ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Distance spectrum ,0101 mathematics ,Barbell graph ,05C12, 05C31, 05C50, 15A15, 15A18, 15B48, 15B57 ,Mathematics ,Doob graph ,Strongly regular graph ,Discrete mathematics ,Distance matrix ,Numerical Analysis ,Optimistic graph ,Algebra and Number Theory ,Resistance distance ,Distance regular graph ,Kneser graph ,Minkowski distance ,Mathematics::Spectral Theory ,Lollipop graph ,010201 computation theory & mathematics ,Odd graph ,Combinatorics (math.CO) ,Geometry and Topology ,MATRIX ,Distance matrices in phylogeny - Abstract
The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs., 20 pages, 3 figures v2. updated acknowledgements
- Published
- 2016
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29. On embeddings of Grassmann graphs in polar Grassmann graphs
- Author
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Mark Pankov
- Subjects
Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,Collinearity ,01 natural sciences ,Linear subspace ,Graph ,Combinatorics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Embedding ,Polar ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Polar coordinate system ,Polar space ,Mathematics - Abstract
We establish that every embedding of a Grassmann graph in a polar Grassmann graph can be reduced to an embedding in a Grassmann graph or to an embedding in the collinearity graph of a polar space. Also, we consider 3-embeddings, i.e. embeddings preserving all distances not greater than 3, of dual polar graphs whose diameter is not less than 3 in polar Grassmann graphs formed by non-maximal singular subspaces. Using the same arguments we show that every such an embedding can be reduced to an embedding in a Grassmann graph.
- Published
- 2016
30. Lower bounds of the skew spectral radii and skew energy of oriented graphs
- Author
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Huishu Lian, Xiaolin Chen, and Xueliang Li
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Spectral radius ,Skew ,05C20, 05C50, 15A18, 05C90 ,Upper and lower bounds ,Graph ,Combinatorics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Adjacency matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let $G$ be a graph with maximum degree $\Delta$, and let $G^{\sigma}$ be an oriented graph of $G$ with skew adjacency matrix $S(G^{\sigma})$. The skew spectral radius $\rho_s(G^{\sigma})$ of $G^\sigma$ is defined as the spectral radius of $S(G^\sigma)$. The skew spectral radius has been studied, but only few results about its lower bound are known. This paper determines some lower bounds of the skew spectral radius, and then studies the oriented graphs whose skew spectral radii attain the lower bound $\sqrt{\Delta}$. Moreover, we apply the skew spectral radius to the skew energy of oriented graphs, which is defined as the sum of the norms of all the eigenvalues of $S(G^\sigma)$, and denoted by $\mathcal{E}_s(G^\sigma)$. As results, we obtain some lower bounds of the skew energy, which improve the known lower bound obtained by Adiga et al., Comment: 16 pages, 2 figures
- Published
- 2015
31. An interlacing approach for bounding the sum of Laplacian eigenvalues of graphs
- Author
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Guillem Perarnau, Willem H. Haemers, Aida Abiad, Miguel Angel Fiol, Research Group: Operations Research, Center Ph. D. Students, and Econometrics and Operations Research
- Subjects
Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,Numerical analysis ,Interlacing ,Mathematics::Spectral Theory ,sum of eigenvalues ,Laplacian eigenvalues ,Graph ,Mathematics - Spectral Theory ,Combinatorics ,Graph spectra ,Bounding overwatch ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,graph spectra ,Geometry and Topology ,Laplacian matrix ,Spectral Theory (math.SP) ,Geometry and topology ,Mathematics - Abstract
We apply interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs. Mainly, we generalize two theorems of Grone and Grone & Merris, providing tight bounds and studying the cases of equality. As a consequence, some results on well-known parameters of a graph, such as the maximum and minimum cuts, the edge-connectivity, the (almost) dominating number, and the edge isoperimetric number, are derived.
- Published
- 2014
32. Spectral moments of trees with given degree sequence
- Author
-
Eric Ould Dadah Andriantiana and Stephan Wagner
- Subjects
Discrete mathematics ,Numerical Analysis ,Spectral moments ,Algebra and Number Theory ,Conjecture ,Degree (graph theory) ,68R05, 05C05, 05C35, 05C50 ,G.2.1 ,G.2.2 ,Mathematics::Spectral Theory ,Graph ,Combinatorics ,Estrada index ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let $\lambda_1,\dots,\lambda_n$ be the eigenvalues of a graph $G$. For any $k\geq 0$, the $k$-th spectral moment of $G$ is defined by $\M_k(G)=\lambda_1^k+\dots+\lambda_n^k$. We use the fact that $\M_k(G)$ is also the number of closed walks of length $k$ in $G$ to show that among trees $T$ whose degree sequence is $D$ or majorized by $D$, $\M_k(T)$ is maximized by the greedy tree with degree sequence $D$ (constructed by assigning the highest degree in $D$ to the root, the second-, third-, \dots highest degrees to the neighbors of the root, and so on) for any $k\geq 0$. Several corollaries follow, in particular a conjecture of Ili\'c and Stevanovi\'c on trees with given maximum degree, which in turn implies a conjecture of Gutman, Furtula, Markovi\'c and Gli\v{s}i\'c on the Estrada index of such trees, which is defined as $\EE(G)=e^{\lambda_1}+\dots+e^{\lambda_n}$., Comment: 24 pages 5 figures
- Published
- 2013
33. Cored hypergraphs, power hypergraphs and their Laplacian H-eigenvalues
- Author
-
Shenglong Hu, Jia-Yu Shao, and Liqun Qi
- Subjects
Numerical Analysis ,Hypergraph ,Mathematics::Combinatorics ,Algebra and Number Theory ,Mathematics::Spectral Theory ,Signless laplacian ,Graph ,Mathematics - Spectral Theory ,Combinatorics ,Computer Science::Discrete Mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Adjacency tensor ,Combinatorics (math.CO) ,Geometry and Topology ,Spectral Theory (math.SP) ,Laplace operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we introduce the class of cored hypergraphs and power hypergraphs, and investigate the properties of their Laplacian H-eigenvalues. From an ordinary graph, one may generate a $k$-uniform hypergraph, called the $k$th power hypergraph of that graph. Power hypergraphs are cored hypergraphs, but not vice versa. Hyperstars, hypercycles, hyperpaths are special cases of power hypergraphs, while sunflowers are a subclass of cored hypergraphs, but not power graphs in general. We show that the largest Laplacian H-eigenvalue of an even-uniform cored hypergraph is equal to its largest signless Laplacian H-eigenvalue. Especially, we find out these largest H-eigenvalues for even-uniform sunflowers. Moreover, we show that the largest Laplacian H-eigenvalue of an odd-uniform sunflower, hypercycle and hyperpath is equal to the maximum degree, i.e., 2. We also compute out the H-spectra of the class of hyperstars. When $k$ is odd, the H-spectra of the hypercycle of size 3 and the hyperpath of length 3 are characterized as well., 27 pages, 3 figures. arXiv admin note: text overlap with arXiv:1304.1315
- Published
- 2013
34. On the second largest eigenvalue of the signless Laplacian
- Author
-
Vladimir Nikiforov and Leonardo Silva de Lima
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Open problem ,010102 general mathematics ,1. No poverty ,010103 numerical & computational mathematics ,Mathematics::Spectral Theory ,Signless laplacian ,01 natural sciences ,Graph ,Combinatorics ,Mathematics - Spectral Theory ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,0101 mathematics ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let $G$ be a graph of order $n,$ and let $q_{1}(G) \geq ...\geq q_{n}(G) $ be the eigenvalues of the $Q$-matrix of $G$, also known as the signless Laplacian of $G.$ In this paper we give a necessary and sufficient condition for the equality $q_{k}(G) =n-2,$ where $1, Comment: This version fills a gap in one proof, noticed by Rundan Xing
- Published
- 2013
- Full Text
- View/download PDF
35. Rank of stably dissipative graphs
- Author
-
Telmo Peixe and Pedro Duarte
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Lotka–Volterra system ,010102 general mathematics ,Reduced graph ,Dynamical Systems (math.DS) ,01 natural sciences ,Graph ,Stably dissipative matrix ,010101 applied mathematics ,Combinatorics ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Dissipative system ,Mathematics - Combinatorics ,Quantitative Biology::Populations and Evolution ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
For the class of stably dissipative Lotka–Volterra systems we prove that the rank of its defining matrix, which is the dimension of the associated invariant foliation, is completely determined by the system’s graph.
- Published
- 2012
- Full Text
- View/download PDF
36. Embeddings of Grassmann graphs
- Author
-
Mark Pankov
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Mathematics - Rings and Algebras ,Semilinear embedding ,Graph ,Combinatorics ,Isometric embedding ,Rings and Algebras (math.RA) ,Grassmann graph ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Mathematics ,Vector space - Abstract
Let V and V ′ be vector spaces of dimension n and n ′ , respectively. Let k ∈ { 2 , … , n - 2 } and k ′ ∈ { 2 , … , n ′ - 2 } . We describe all isometric and l -rigid isometric embeddings of the Grassmann graph Γ k ( V ) in the Grassmann graph Γ k ′ ( V ′ ) .
- Published
- 2012
37. The energy of random graphs
- Author
-
Du, Wenxue, Li, Xueliang, and Li, Yiyang
- Subjects
Numerical Analysis ,Mathematics::Combinatorics ,15A18 ,Algebra and Number Theory ,05C80 ,05C90 ,15A52 ,Eigenvalues ,Random matrix ,Mathematics - Statistics Theory ,Random graph ,Statistics Theory (math.ST) ,92E10 ,Limiting spectral distribution ,Graph ,Graph energy ,Empirical spectral distribution ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology - Abstract
In 1970s, Gutman introduced the concept of the energy $\En(G)$ for a simple graph $G$, which is defined as the sum of the absolute values of the eigenvalues of $G$. This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among which bipartite graphs are of particular interest. But there are only a few graphs attaining the equalities of those bounds. We however obtain an exact estimate of the energy for almost all graphs by Wigner's semi-circle law, which generalizes a result of Nikiforov. We further investigate the energy of random multipartite graphs by considering a generalization of Wigner matrix, and obtain some estimates of the energy for random multipartite graphs., 16 pages
- Published
- 2011
38. Choice number and energy of graphs
- Author
-
Ebrahim Ghorbani and Saieed Akbari
- Subjects
Numerical Analysis ,Energy ,Algebra and Number Theory ,Graph ,05C15, 05C50, 15A03 ,Choice number ,Combinatorics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Adjacency matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved that E(G)>= 2(n-��(\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>= 2ch(G) for all graphs G except for those in a few specified families, where \bar{G}, ��(G), and ch(G) are the complement, the chromatic number, and the choice number of G, respectively., to appear in Linear Algebra and its Applications
- Published
- 2008
39. Eigenvalues and degree deviation in graphs
- Author
-
Vladimir Nikiforov
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Degree (graph theory) ,Graph theory ,Graph ,15A42, 05C50 ,Combinatorics ,Constant factor ,Degree sequence ,Graph eigenvalues ,FOS: Mathematics ,Bipartite graph ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Semiregular graph ,Combinatorics (math.CO) ,Adjacency matrix ,Geometry and Topology ,Measure of irregularity ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let G be a graph with n vertices and m edges and let μ(G) = μ1(G) ⩾ ⋯ ⩾ μn(G) be the eigenvalues of its adjacency matrix. Set s ( G ) = ∑ u ∈ V ( G ) ∣ d ( u ) - 2 m / n ∣ . We prove that s 2 ( G ) 2 n 2 2 m ⩽ μ ( G ) - 2 m n ⩽ s ( G ) . In addition we derive similar inequalities for bipartite G. We also prove that the inequality μ k ( G ) + μ n - k + 2 ( G ¯ ) ⩾ - 1 - 2 2 s ( G ) holds for every k = 2, … , n. Finally we prove that for every graph G of order n, μ n ( G ) + μ n ( G ¯ ) ⩽ - 1 - s 2 ( G ) 2 n 3 . We show that these inequalities are tight up to a constant factor.
- Published
- 2006
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- View/download PDF
40. An inequality involving the second largest and smallest eigenvalue of a distance-regular graph
- Author
-
Jack H. Koolen, Hyonju Yu, and Jongyook Park
- Subjects
Discrete mathematics ,Shill distance-regular graphs ,Numerical Analysis ,Algebra and Number Theory ,Inequality ,media_common.quotation_subject ,Mathematics::Spectral Theory ,Distance-regular graph ,Graph ,Combinatorics ,Tight distance-regular graph ,FOS: Mathematics ,Rayleigh–Faber–Krahn inequality ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Regular graph ,Combinatorics (math.CO) ,Geometry and Topology ,Bounds on eigenvalues ,Eigenvalues and eigenvectors ,Computer Science::Information Theory ,05E30 ,Mathematics ,media_common - Abstract
For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1), 15 pages, this is submitted to Linear Algebra and Applications.
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41. Perfect state transfer in cubelike graphs
- Author
-
Wang Chi Cheung and Chris Godsil
- Subjects
Vertex (graph theory) ,Quantum Physics ,Numerical Analysis ,Algebra and Number Theory ,Code word ,FOS: Physical sciences ,Graph theory ,Graph ,Combinatorics ,Binary codes ,Cubelike graph ,Perfect state transfer ,Greatest common divisor ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Binary code ,Geometry and Topology ,Combinatorics (math.CO) ,Quantum Physics (quant-ph) ,Mathematics ,Vector space - Abstract
Suppose $C$ is a subset of non-zero vectors from the vector space $\mathbb{Z}_2^d$. The cubelike graph $X(C)$ has $\mathbb{Z}_2^d$ as its vertex set, and two elements of $\mathbb{Z}_2^d$ are adjacent if their difference is in $C$. If $M$ is the $d\times |C|$ matrix with the elements of $C$ as its columns, we call the row space of $M$ the code of $X$. We use this code to study perfect state transfer on cubelike graphs. Bernasconi et al have shown that perfect state transfer occurs on $X(C)$ at time $\pi/2$ if and only if the sum of the elements of $C$ is not zero. Here we consider what happens when this sum is zero. We prove that if perfect state transfer occurs on a cubelike graph, then it must take place at time $\tau=\pi/2D$, where $D$ is the greatest common divisor of the weights of the code words. We show that perfect state transfer occurs at time $\pi/4$ if and only if D=2 and the code is self-orthogonal., Comment: 10 pages, minor revisions
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42. Expected values of parameters associated with the minimum rank of a graph
- Author
-
H. Tracy Hall, Leslie Hogben, Bryan L. Shader, and Ryan Martin
- Subjects
05C50, 05C80, 15A03 ,Average maximum nullity ,Average minimum rank ,Random graph ,Expected value ,Maximum nullity ,Graph ,Combinatorics ,Positive semidefinite minimum rank ,Random regular graph ,Colin de Verdière type parameter ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Mathematics ,Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,Matrix ,Minimum rank of a graph ,Minimum rank ,Delta conjecture ,Rank ,Bound graph ,Combinatorics (math.CO) ,Geometry and Topology ,Random variable - Abstract
We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdi\`ere-type parameters. Let $G(v,p)$ denote the usual Erd\H{o}s-R\'enyi random graph on $v$ vertices with edge probability $p$. We obtain bounds for the expected value of the random variables ${\rm mr}(G(v,p))$, ${\rm M}(G(v,p))$, $\nu(G(v,p))$ and $\xi(G(v,p))$, which yield bounds on the average values of these parameters over all labeled graphs of order $v$., Comment: 17 pages, 2 figures
- Full Text
- View/download PDF
43. On the maximal energy tree with two maximum degree vertices
- Author
-
Xueliang Li, Jing Li, and Yongtang Shi
- Subjects
Numerical Analysis ,Simple graph ,Algebra and Number Theory ,Coulson integral formula ,Mathematics::Geometric Topology ,Graph ,Vertex (geometry) ,Graph energy ,Combinatorics ,Maximal energy ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Turning point ,Geometry and Topology ,Combinatorics (math.CO) ,05C50, 05C90, 15A18, 92E10 ,Eigenvalues and eigenvectors ,Tree ,Mathematics - Abstract
For a simple graph $G$, the energy $E(G)$ is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix. For $\Delta\geq 3$ and $t\geq 3$, denote by $T_a(\Delta,t)$ (or simply $T_a$) the tree formed from a path $P_t$ on $t$ vertices by attaching $\Delta-1$ $P_2$'s on each end of the path $P_t$, and $T_b(\Delta, t)$ (or simply $T_b$) the tree formed from $P_{t+2}$ by attaching $\Delta-1$ $P_2$'s on an end of the $P_{t+2}$ and $\Delta -2$ $P_2$'s on the vertex next to the end. In [X. Li, X. Yao, J. Zhang and I. Gutman, Maximum energy trees with two maximum degree vertices, J. Math. Chem. 45(2009), 962--973], Li et al. proved that among trees of order $n$ with two vertices of maximum degree $\Delta$, the maximal energy tree is either the graph $T_a$ or the graph $T_b$, where $t=n+4-4\Delta\geq 3$. However, they could not determine which one of $T_a$ and $T_b$ is the maximal energy tree. This is because the quasi-order method is invalid for comparing their energies. In this paper, we use a new method to determine the maximal energy tree. It turns out that things are more complicated. We prove that the maximal energy tree is $T_b$ for $\Delta\geq 7$ and any $t\geq 3$, while the maximal energy tree is $T_a$ for $\Delta=3$ and any $t\geq 3$. Moreover, for $\Delta=4$, the maximal energy tree is $T_a$ for all $t\geq 3$ but $t=4$, for which $T_b$ is the maximal energy tree. For $\Delta=5$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t$ is odd and $3\leq t\leq 89$, for which $T_a$ is the maximal energy tree. For $\Delta=6$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t=3,5,7$, for which $T_a$ is the maximal energy tree. One can see that for most $\Delta$, $T_b$ is the maximal energy tree, $\Delta=5$ is a turning point, and $\Delta=3$ and 4 are exceptional cases., Comment: 16 pages
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- View/download PDF
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