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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.

Let Gbe a simple connected graph of order n, D(G) the distance matrix of G, and Tr(G) the diagonal matrix of the vertex transmissions of G. The generalized distance matrix of Gis defined as Dα(G)=αTr(G)+(1-α)D(G), 0≤α≤1. It is also commonly referred to as the Dα-matrix of G. Previously, some scholars have studied the positive semidefinite properties of Dα-matrix of some special graphs. In this paper, we will study the positive semidefinite properties of Dα-matrix of four special graphs, and give the corresponding conditions that αneeds to satisfy.

For a simple connected graph G, the Q-generating function of the numbers Nkof semi-edge walks of length kin Gis defined by WQ(t)=∑k=0∞Nktk. This paper reveals that the Q-generating function WQ(t)may be expressed in terms of the Q-polynomials of the graph Gand 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 WQ(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 (λIn-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.

A mixed graph MGis the graph obtained from an unoriented simple graph Gby giving directions to some edges of G, where Gis often called the underlying graph of MG. In this paper, we discuss the determinants of two classes of incidence matrices of the second kind of rootless mixed trees and unicyclic mixed graphs. Applying these results, we characterize the explicit expressions of various minors for Hermitian (quasi-)Laplacian matrix of the second kind of MG. Moreover, we give two sufficient conditions that the module of any cofactor of Hermitian (quasi-)Laplacian matrix of the second kind of the mixed graph MGis equal to the number of spanning trees of the underlying graph G.

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.

Given two graphs G1of order n1and G2, the neighborhood corona of G1and G2, denoted by G1★G2, is the graph obtained by taking one copy of G1and taking n1copies of G2, in the meanwhile, linking all the neighbors of the i-th vertex of G1with all vertices of the i-th copy of G2. In our work, we give some conditions that G1★G2is not periodic. Furthermore, we demonstrate some sufficient conditions for G1★G2having no perfect state transfer. Some examples are provided to explain our results. In addition, for the reason that the graph admitting perfect state transfer is rare, we also consider pretty good state transfer on neighborhood corona of two graphs. We show some sufficient conditions for G1★G2admitting pretty good state transfer.