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The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. Recently, there has been a renewed interest in studying Seidel spectrum of graphs and some achievements have been made in this regard. In this paper, we determine the Seidel spectra of some variants of corona operations, such as edge corona, subdivision-vertex corona, subdivision-vertex neighbourhood corona of two graphs and so on. The corresponding Seidel eigenvectors are also described completely. Applying the obtained results, we give some sufficient and necessary conditions for edge corona, subdivision-vertex corona and subdivision-vertex neighbourhood corona of two graphs to be Seidel integral. [ABSTRACT FROM AUTHOR]

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. For two simple connected graphs G 1 and G 2 , let us denote the subdivision-vertex join and subdivision-edge join by G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 , respectively. In this paper, we completely determine the Seidel spectrum and corresponding Seidel eigenvectors of G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 . As an application, we give a sufficient and necessary condition for G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 to be Seidel integral. [ABSTRACT FROM AUTHOR]

Let S(G) be the Seidel matrix of a graph G. The Seidel energy of G, denoted by ES(G), is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix S(G) of G. In this paper, we consider the change of Seidel energy of graphs due to edge deletion. It is proved that the Seidel energy of the tripartite Turán graph T(n, 3) of order n ≥ 9 is always increased when an edge is deleted. [ABSTRACT FROM AUTHOR]

Given any real α ∈ [ 0 , 1 ] , the α -index of a graph G is the largest eigenvalue λ α (G) of the matrix A α (G) = α D (G) + (1 − α) A (G) , where A (G) and D (G) stand for the adjacency matrix and the diagonal matrix of vertex degrees in G , respectively. In this paper, we determine the unique graph with maximum α -index among all graphs of order n with no 4-cycle and 5-cycle, respectively. These results partially answer a proposed problem in Nikiforov (2017) [7, Problem 24]. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with vertex set V (G) = { v 1 , v 2 , ... , v n } and edge set E (G). The arithmetic–geometric matrix A AG (G) = (h i j) of G is a matrix of order n defined by h i , j = 1 2 d i d j + d j d i if v i v j ∈ E (G) and 0 otherwise, where d i stands for the degree of the vertex v i in G. The arithmetic–geometric characteristic polynomial of G is the characteristic polynomial of A AG (G). The arithmetic–geometric energy ℰ AG (G) of G is the sum of absolute values of all eigenvalues of A AG (G). In this paper, we obtain the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy of some specific graphs. In addition, we also consider the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy change of these graphs when one edge is deleted. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n with adjacency matrix A (G) and degree diagonal matrix D (G). The A α -matrix of G is defined as A α (G) = α D (G) + (1 − α) A (G) for any α ∈ [ 0 , 1). All the eigenvalues of A α (G) are called A α -eigenvalues of G , and the A α -spectrum of G consists of all the A α -eigenvalues along with their multiplicities. Denote the set of connected graphs (resp., of order n) with distinct A α -eigenvalues by G A α (resp., G n A α ). We also use G A α ⁎ (resp., G n A α ⁎ ) to denote the set of connected graphs in G A α (resp., G n A α ) whose A α -eigenvalues are main. Two graphs G and H are called A α -cospectral if they have the same A α -spectrum. This paper proposes a new approach to construct infinite families of G A α and G A α ⁎ . More specifically, for a graph G in G n A α or G n A α ⁎ , the infinite families of G A α or G A α ⁎ are constructed from G. At the same time, the A α -spectra of these graphs are completely determined by the A α -spectrum of G. Finally, using this technique, we also construct some infinite families of non-isomorphic A α -cospectral graphs in G A α and G A α ⁎ . Applying the results obtained to two cases of adjacency matrix and signless Laplacian matrix, we can derive the main results in [9] and [10]. [ABSTRACT FROM AUTHOR]

Many variants of join operations of graphs have been introduced, and their spectral properties have been studied extensively by many researchers. This paper mainly focuses on the Laplacian spectra of some double join operations of graphs. We first introduce the conception of double join matrix and provide a complete information about its eigenvalues and the corresponding eigenvectors. Further, we define four variants of double join operations based on subdivision graph, Q-graph, R-graph and total graph. Applying the result obtained about double join matrices, we give an explicit complete characterization of the Laplacian eigenvalues and the corresponding eigenvectors of four variants in terms of the Laplacian eigenvalues and the eigenvectors of factor graphs. These results generalize some well-known results on the Laplacian spectra of some join operations of graphs. [ABSTRACT FROM AUTHOR]

Abstract: The energy of a digraph D is defined as , where are the (possibly complex) eigenvalues of D. In this paper, we first give an improved lower bound on the spectral radius of the digraph D. Using this result, we obtain a new upper bound on the energy and characterize some extreme digraphs which attain this upper bound. This result theoretically improves and generalizes some known results. [Copyright &y& Elsevier]

We establish an inclusion relation between two known inclusion intervals of matrix singular values in some special case. In addition, based on the use of positive scale vectors, a known inclusion interval of matrix singular values is also improved. [ABSTRACT FROM AUTHOR]

Abstract: Let D be a digraph of order n and denote all the eigenvalues of the skew-adjacency matrix of D. The skew energy of D is defined as . In this paper, it is proved that for any positive integer , there exists a k-regular graph of order n having an orientation D with . This work positively answers a problem proposed by Adiga et al. [C. Adiga, R. Balakrishnan, Wasin So, The skew energy of a digraph, Linear Algebra Appl. 432 (2010) 1825–1835]. In addition, a digraph is also constructed such that its skew energy is the same as the energy of its underlying graph. [Copyright &y& Elsevier]

Abstract: For a graph and a real number , the graph invariant is the sum of the th power of the non-zero Laplacian eigenvalues of . In this note, we obtain some bounds of for a connected bipartite graph , which improve some known results of Zhou [B. Zhou, On sum of powers of the Laplacian eigenvalues of graphs, Linear Algebra Appl. 429 (2008) 2239-2246]. [Copyright &y& Elsevier]

For a simple graph G , its Q -graph Q (G) is derived from G by adding one new point in every edge of G and linking two new vertices by edge if they are between two edges that having a common endpoint. In our work, we demonstrate that for a regular graph G , if all the signless Laplacian eigenvalues are integers, then the Q (G) exists no signless Laplacian perfect state transfer. We also present a sufficient restriction that the Q (G) admits signless Laplacian pretty good state transfer when G exhibits signless Laplacian perfect state transfer between two specific vertices for a regular graph G. In addition, in view of these results, we also present some new families of Q -graphs, which have no signless Laplacian perfect state transfer, but admit signless Laplacian pretty good state transfer. [ABSTRACT FROM AUTHOR]