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Some relations between rank, chromatic number and energy of graphs
, Article Discrete Mathematics ; Volume 309, Issue 3 , 2009 , Pages 601-605 ; 0012365X (ISSN) ; Ghorbani, E ; Zare, S ; Sharif University of Technology
2009
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. Let G be a graph of order n and rank (G) be the rank of the adjacency matrix of G. In this paper we characterize all graphs with E (G) = rank (G). Among other results we show that apart from a few families of graphs, E (G) ≥ 2 max (χ (G), n - χ (over(G, -))), where n is the number of vertices of G, over(G, -) and χ (G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E (G) in terms of rank (G) are given. © 2008 Elsevier B.V. All rights reserved
Edge addition, singular values, and energy of graphs and matrices
, Article Linear Algebra and Its Applications ; Volume 430, Issue 8-9 , 2009 , Pages 2192-2199 ; 00243795 (ISSN) ; Ghorbani, E ; Oboudi, M. R ; Sharif University of Technology
2009
Abstract
The energy of a graph/matrix is the sum of the absolute values of its eigenvalues. We investigate the result of duplicating/removing an edge to the energy of a graph. We also deal with the problem that which graphs G have the property that if the edges of G are covered by some subgraphs, then the energy of G does not exceed the sum of the subgraphs' energies. The problems are addressed in the general setting of energy of matrices which leads us to consider the singular values too. Among the other results it is shown that the energy of a complete multipartite graph increases if a new edge added or an old edge is deleted. © 2008 Elsevier Inc. All rights reserved
On the minimum energy of regular graphs
, Article Linear Algebra and Its Applications ; Volume 581 , 2019 , Pages 51-71 ; 00243795 (ISSN) ; Akbari, S ; Ghasemian, E ; Ghodrati, A. H ; Hosseinzadeh, M. A ; Koorepazan Moftakhar, F ; Sharif University of Technology
Elsevier Inc
2019
Abstract
The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E(G)≥n. Here, we improve this result by showing that if G is a connected subcubic graph of order n≥8, then E(G)≥1.01n. Also, we prove that if G is a traceable subcubic graph of order n≥8, then E(G)>1.1n. Let G be a connected cubic graph of order n≥8, it is shown that E(G)>n+2. It was proved that if G is a connected cubic graph of order n, then E(G)≤1.65n. Also, in this paper we would like to present the best lower bound for the energy of a connected cubic graph. We introduce an infinite family of connected cubic graphs whose for...
Choice number and energy of graphs
, Article Linear Algebra and Its Applications ; Volume 429, Issue 11-12 , 2008 , Pages 2687-2690 ; 00243795 (ISSN) ; Ghorbani, E ; Sharif University of Technology
2008
Abstract
The energy of a graph G, denoted by E (G), is defined as the sum of the absolute values of all eigenvalues of the adjacency matrix of G. It is proved that E (G) ≥ 2 (n - χ (over(G, -))) ≥ 2 (ch (G) - 1) for every graph G of order n, and that E (G) ≥ 2 ch (G) for all graphs G except for those in a few specified families, where over(G, -), χ (G), and ch (G) are the complement, the chromatic number, and the choice number of G, respectively. © 2007 Elsevier Inc. All rights reserved
Hypoenergetic and nonhypoenergetic digraphs
, Article Linear Algebra and Its Applications ; Volume 618 , 2021 , Pages 129-143 ; 00243795 (ISSN) ; Das, K. C ; Khalashi Ghezelahmad, S ; Koorepazan Moftakhar, F ; Sharif University of Technology
Elsevier Inc
2021
Abstract
The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by Nikiforov to arbitrary complex matrices. Recall that the trace norm of a digraph D is defined as, N(D)=∑i=1nσi, where σ1≥⋯≥σn are the singular values of the adjacency matrix of D. In this paper we would like to present some lower and upper bounds for N(D). For any digraph D it is proved that N(D)≥rank(D) and the equality holds if and only if D is a disjoint union of directed cycles and directed paths. Finally, we present a lower bound on σ1 and N(D) in terms of the size of digraph D. © 2021 Elsevier Inc
Hypoenergetic and nonhypoenergetic digraphs
, Article Linear Algebra and Its Applications ; Volume 618 , 2021 , Pages 129-143 ; 00243795 (ISSN) ; Das, K. C ; Khalashi Ghezelahmad, S ; Koorepazan Moftakhar, F ; Sharif University of Technology
Elsevier Inc
2021
Abstract
The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by Nikiforov to arbitrary complex matrices. Recall that the trace norm of a digraph D is defined as, N(D)=∑i=1nσi, where σ1≥⋯≥σn are the singular values of the adjacency matrix of D. In this paper we would like to present some lower and upper bounds for N(D). For any digraph D it is proved that N(D)≥rank(D) and the equality holds if and only if D is a disjoint union of directed cycles and directed paths. Finally, we present a lower bound on σ1 and N(D) in terms of the size of digraph D. © 2021 Elsevier Inc
Some lower bounds for the energy of graphs
, Article Linear Algebra and Its Applications ; Volume 591 , 2020 , Pages 205-214 ; Ghodrati, A. H ; Hosseinzadeh, M. A ; Sharif University of Technology
Elsevier Inc
2020
Abstract
The singular values of a matrix A are defined as the square roots of the eigenvalues of A⁎A, and the energy of A denoted by E(A) is the sum of its singular values. The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A=(BDD⁎C), then E(A)≥2E(D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E(H) is an edge cut of G, then E(H)≤E(G), i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known...
On the energy of line graphs
, Article Linear Algebra and Its Applications ; Volume 636 , 2022 , Pages 143-153 ; 00243795 (ISSN) ; Alazemi, A ; Anđelić, M ; Hosseinzadeh, M. A ; Sharif University of Technology
Elsevier Inc
2022
Abstract
The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In Akbari and Hosseinzadeh (2020) [3] it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree δ(G) whose adjacency matrix is non-singular, E(G)≥Δ(G)+δ(G) and the equality holds if and only if G is a complete graph. Let G be a connected graph with the edge set E(G). In this paper, first we show that E(L(G))≥|E(G)|+Δ(G)−5, where L(G) denotes the line graph of G. Next, using this result, we prove the validity of the conjecture for the line of each connected graph of order at least 7. © 2021 Elsevier Inc