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Let G be a graph. The energy of G, (Formula presented.), is defined as the sum of absolute values of its eigenvalues. Here, it is shown that if G is a graph and (Formula presented.) is an edge partition of G, such that (Formula presented.) are spanning; then (Formula presented.) if and only if (Formula presented.), for every (Formula presented.) and (Formula presented.), where (Formula presented.) is the adjacency matrix of (Formula presented.). It was proved that if G is a graph and (Formula presented.) are subgraphs of G which partition edges of G, then (Formula presented.). In this paper we show that if G is connected, then the equality is strict, that is (Formula presented.). © 2022... 

Let G and H be two graphs. A proper vertex coloring of G is called a dynamic coloring, if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic coloring with k colors denoted by χ2(G). We denote the cartesian product of G and H by G□H. In this paper, we prove that if G and H are two graphs and δ(G) ≥ 2, then χ2(G□H) ≤ max(χ2(G),x(H)). We show that for every two natural numbers m and n, m,n ≥ 2, χ2(Pm□Pn) = 4. Also, among other results it is shown that if 3|mn, then χ2(C m□Cn) = 3 and otherwise χ2(C m□Cn) = 4  

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group  

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group  

Let G be a simple graph with n vertices and let λ1, λ2,...,λn be its eigenvalues. The energy of G is defined to be E(G) = ∑i=1n|λi|. In this note, for a given k-regular graph we find explicit formulas for the energy of L(G), the complement of line graph of G. This provides us with some practical ways to compute the energy of a large family of regular graphs  

The energy E of any n-vertex regular graph G of degree r, r > 0, is greater than or equal to n. Equality holds if and only if every component of G is isomorphic to the complete bipartite graph Kr,r. If G is triangle- and quadrangle-free, then E ≥ nr/√2r - 1. In particular, for any fullerene and nanotube with n carbon atoms, 1.34n ≤ E ≤ 1.73 n  

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraph Knr is decomposable into commuting perfect matchings if and only if n is a 2-power. Also, it is shown that the complete graph Kn is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 2006 Wiley Periodicals, Inc  

For a graph G, let (Formula presented.) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019 Informa UK Limited, trading as Taylor & Francis Group  

The energy of a graph G, ϵ(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number µ(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ∆ ≥ 6 we present two new upper bounds for the energy of a graph: (Formula presented) and (Formula presented). The latter one improves recently obtained bound (Formula presented) where ∆e stands for the largest edge degree and a = 2(∆e + 1). We also present a short proof of this result and several open problems. © 2021  

For a graph G, let (Formula presented.) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019 Informa UK Limited, trading as Taylor & Francis Group  

The energy of a graph G, ϵ(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number µ(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ∆ ≥ 6 we present two new upper bounds for the energy of a graph: (Formula presented) and (Formula presented). The latter one improves recently obtained bound (Formula presented) where ∆e stands for the largest edge degree and a = 2(∆e + 1). We also present a short proof of this result and several open problems. © 2021  

The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all non-trivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I⊆J or J⊆I. In this paper, we show that In(R) is not connected if and only if R≅M2(D) or D1×D2, for some division rings, D, D1 and D2. Moreover, if R is connected, then diam(In(R))≤3. We prove that if In(R) is a tree, then In(R) is a star graph or P4. Also, In(R) is a complete graph if and only if R is a uniserial ring. Next, it is shown that the inclusion ideal graph of Mn(D) for a division ring D and a natural number n>3 is not regular  

The minimum number of complete bipartite subgraphs needed to partition the edges of a graph G is denoted by b (G). A known lower bound on b (G) states that b (G) ≥max {p (G), q (G)}, where p (G) and q (G) are the numbers of positive and negative eigenvalues of the adjacency matrix of G, respectively. When equality is attained, G is said to be eigensharp and when b (G) = max {p (G), q (G)} + 1, G is called an almost eigensharp graph. In this paper, we investigate the eigensharpness of graphs with at most one cycle and products of some families of graphs. Among the other results, we show that Pm ∨ Pn, Cm ∨ Pn for m ≡ 2, 3 (mod 4) and Qn when n is odd are eigensharp. We obtain some results on... 

A Deza graph with parameters (n, k, b, a) is a k-regular graph with n vertices such that any two of its vertices have b or a common neighbours, where b ≥ a. In this paper we investigate spectra of Deza graphs. In particular, using the eigenvalues of a Deza graph we determine the eigenvalues of its children. Divisible design graphs are significant cases of Deza graphs. Sufficient conditions for Deza graphs to be divisible design graphs are given, a few families of divisible design graphs are presented and their properties are studied. Our special attention goes to the invertibility of the adjacency matrices of Deza graphs. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group  

A Deza graph with parameters (Formula presented.) is a k-regular graph with n vertices such that any two of its vertices have b or a common neighbours, where (Formula presented.). In this paper we investigate spectra of Deza graphs. In particular, using the eigenvalues of a Deza graph we determine the eigenvalues of its children. Divisible design graphs are significant cases of Deza graphs. Sufficient conditions for Deza graphs to be divisible design graphs are given, a few families of divisible design graphs are presented and their properties are studied. Our special attention goes to the invertibility of the adjacency matrices of Deza graphs. © 2020 Informa UK Limited, trading as Taylor &... 

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... 

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  

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ2(G)+λ2(G‾)≥1, where G‾ is the complement of G. In this paper, it is shown that max⁡{λ2(G),λ2(G‾)}≥[Formula presented]. © 2018 Elsevier Inc  

For a graph G of order n, 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. For any x∈R n , let ∇ x ∈R (n2) be the vector whose {i,j}-th entry is |x i −x j |. In this paper, we show the aforementioned conjecture is equivalent to prove that every two orthonormal vectors f,g∈R n with zero mean satisfy ‖∇ f −∇ g ‖ 2 ≥2. In this article, it is shown that for the validity of the conjecture it suffices to prove that the conjecture holds for all permutation graphs. © 2019 Elsevier Inc  

In this article we introduce the cylindrical construction, as an edge-replacement procedure admitting twists on both ends of the hyperedges, generalizing the concepts of lifts and Pultr templates at the same time. We prove a tensor-hom duality for this construction and we show that not only a large number of well-known graph constructions are cylindrical but also the construction and its dual give rise to some new graph constructions, applications and results. To show the applicability of the main duality we introduce generalized Grötzsch, generalized Petersen-like and Coxeter-like graphs and we prove some coloring properties of these graphs