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Given a simple graph G, its Laplacian-energy-like invariant LEL(G) and incidence energy IE(G) are the sum of square root of its all Laplacian eigenvalues and signless Laplacian eigenvalues, respectively. This paper obtains some improved bounds on LEL and IE of the 𝓡-graph and 𝓠-graph for a regular graph. Theoretical analysis indicates that these results improve some known results. In addition, some new lower bounds on LEL and IE of the line graph of a semiregular graph are also given.

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]

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 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]