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Abstract Let D (G) and D i a g (T r) denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph G , respectively. The distance signless Laplacian matrix of G is defined as D Q (G) = D i a g (T r) + D (G). Heretofore, the spectral properties of D (G) and D Q (G) have attracted much more attention. In the present paper, we propose to study the convex combinations D α (G) of D i a g (T r) and D (G) , defined as D α (G) = α D i a g (T r) + (1 − α) D (G) , 0 ≤ α ≤ 1. This study sheds new light on D (G) and D Q (G). Some spectral properties of D α (G) are given and a few open problems are discussed. Furthermore, we take effort to obtain some upper and lower bounds of spectral radius of D α (G). Finally, the generalized distance spectra of some graphs obtained by operations are also studied. [ABSTRACT FROM AUTHOR]

Let G [ F, V k , H v ] be the graph with k pockets, where F is a simple graph of order n  ≥ 1, V k = { v 1 , … , v k } is a subset of the vertex set of F and H v is a simple graph of order m  ≥ 2, v is a specified vertex of H v . Also let G [ F, E k , H uv ] be the graph with k edge-pockets, where F is a simple graph of order n  ≥ 2, E k = { e 1 , … , e k } is a subset of the edge set of F and H uv is a simple graph of order m  ≥ 3, uv is a specified edge of H uv such that H u v − u is isomorphic to H u v − v . In this paper, we obtain some results describing the signless Laplacian spectra of G [ F, V k , H v ] and G [ F, E k , H uv ] in terms of the signless Laplacian spectra of F, H v and F, H uv , respectively. In addition, we also give some results describing the adjacency spectrum of G [ F, V k , H v ] in terms of the adjacency spectra of F, H v . Finally, as many applications of these results, we construct infinitely many pairs of signless Laplacian (resp. adjacency) cospectral graphs. [ABSTRACT FROM AUTHOR]

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]

Abstract: Let G be a simple connected graph of order n with degree sequence in non-increasing order. The signless Laplacian spectral radius of G is the largest eigenvalue of its signless Laplacian matrix . In this paper, we give a sharp upper bound on the signless Laplacian spectral radius in terms of , which improves and generalizes some known results. [Copyright &y& Elsevier]

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]

Abstract: Let be two simple connected graphs. Denote the corona and the edge corona of by and , respectively. In this paper, we first introduce a new invariant, the -coronal of a graph matrix , where the matrix is associated with a graph in a prescribed way. Then it is used to compute the signless Laplacian spectrum of and in terms of the signless Laplacian spectrum of and . In addition, the spectrum of are also given in terms of the spectrum of and . Finally, as an application of these results, we construct many pairs of nonisomorphic signless Laplacian cospectral graphs. [Copyright &y& Elsevier]

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]