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For a simple connected graph G, the Q-generating function of the numbers N k of semi-edge walks of length k in G is defined by W Q (t) = ∑ k = 0 ∞ N k t k . This paper reveals that the Q-generating function W Q (t) may be expressed in terms of the Q-polynomials of the graph G and its complement G ¯ . Using this result, we study some Q-spectral properties of graphs and compute the Q-polynomials for some graphs obtained from various graph operations, such as the complement graph of a regular graph, the join of two graphs and the (edge)corona of two graphs. As another application of the Q-generating function W Q (t) , we also give a combinatorial interpretation of the Q-coronal of G, which is defined to be the sum of the entries of the matrix (λ I n - Q (G)) - 1 . This result may be used to obtain the many alternative calculations of the Q-polynomials of the (edge)corona of two graphs. Further, we also compute the Q-generating functions of the join of two graphs and the complete multipartite graphs. [ABSTRACT FROM AUTHOR]

For two graphs G and H, the corona product G ∘ H is the graph obtained by taking one copy of G and | V G | copies of H, and joining the ith vertex of G with every vertex of the ith copy of H. In this paper, we study the state transfer of corona relative to the signless Laplacian matrix. We explore some conditions that guarantee the signless Laplacian perfect state transfer in G ∘ H . We prove that G ∘ K m has no signless Laplacian perfect state transfer for some special m. We also show that K 2 ∘ H has pretty good state transfer but no perfect state transfer relative to the signless Laplacian matrix for a regular graph H. Furthermore, we show that n K 2 ¯ ∘ K 1 has signless Laplacian pretty good state transfer, where n K 2 ¯ is the cocktail party graph. [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 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]

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]