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Given a graph $G$ on $n$ vertices, its adjacency matrix and degree diagonal matrix are represented by $A(G)$ and $D(G)$, respectively. The $Q$-spectrum of $G$ consists of all the eigenvalues of its signless Laplacian matrix $Q(G)=A(G)+D(G)$ (including the multiplicities). A graph $G$ is known as being determined by its generalized $Q$-spectrum ($DGQS$ for short) if, for any graph $H$, $H$ and $G$ have the same $Q$-spectrum and so do their complements, then $H$ is isomorphic to $G$. In this paper, we present a method to construct $DGQS$ graphs. More specifically, let the matrix $W_{Q}(G)=\left [e,Qe,\dots ,Q^{n-1}e \right ]$ ($e$ denotes the all-one column vector ) be the $Q$-walk matrix of $G$. It is shown that $G\circ P_{k}$ ($k=2,3$) is $DGQS$ if and only if $G$ is $DGQS$ for some specific graphs. This also provides a way to construct $DGQS$ graphs with more vertices by using $DGQS$ graphs with fewer vertices. At the same time, we also prove that $G\circ P_{2}$ is still $DGQS$ under specific circumstances. In particular, on the basis of the above results, we obtain an infinite sequences of $DGQS$ graphs $G\circ P_{k}^{t}$ ($k=2,3;t\ge 1$) for some specific $DGQS$ graph $G$., Comment: 15 pages, 1 figures

A mixed graph $M_{G}$ is the graph obtained from an unoriented simple graph $G$ by giving directions to some edges of $G$, where $G$ is often called the underlying graph of $M_{G}$. In this paper, we introduce two classes of incidence matrices of the second kind of $M_{G}$, and discuss the determinants of these two matrices for 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 $M_{G}$. Moreover, we give two sufficient conditions that the absolute values of all the cofactors of Hermitian (quasi-)Laplacian matrix of the second kind are equal to the number of spanning trees of the underlying graph $G$., Comment: 16 pages,7 figures

Given two graphs $G_{1}$ of order $n_{1}$ and $G_{2}$, the neighborhood corona of $G_{1}$ and $G_{2}$, denoted by $G_{1}\bigstar G_{2}$, is the graph obtained by taking one copy of $G_{1}$ and taking $n_{1}$ copies of $G_{2}$, in the meanwhile, linking all the neighbors of the $i$-th vertex of $G_{1}$ with all vertices of the $i$-th copy of $G_{2}$. In our work, we give some conditions that $G_{1}\bigstar G_{2}$ is not periodic. Furthermore, we demonstrate some sufficient conditions for $G_{1}\bigstar G_{2}$ having 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 $G_{1}\bigstar G_{2}$ admitting pretty good state transfer., Comment: 17 pages

Let $G$ be a simple undirected connected graph with the Harary matrix $RD(G)$, which is also called the reciprocal distance matrix of $G$. The reciprocal distance signless Laplacian matrix of $G$ is $RQ(G)=RT(G)+RD(G)$, where $RT(G)$ denotes the diagonal matrix of the vertex reciprocal transmissions of graph $G$. This paper intends to introduce a new matrix $RD_{\alpha}(G)=\alpha RT(G)+(1-\alpha)RD(G)$, $\alpha\in [0,1]$, to track the gradual change from $RD(G)$ to $RQ(G)$. First, we describe completely the eigenvalues of $RD_{\alpha}(G)$ of some special graphs. Then we obtain serval basic properties of $RD_{\alpha}(G)$ including inequalities that involve the spectral radii of the reciprocal distance matrix, reciprocal distance signless Laplacian matrix and $RD_{\alpha}$-matrix of $G$. We also provide some lower and upper bounds of the spectral radius of $RD_{\alpha}$-matrix. Finally, we depict the extremal graphs with maximal spectral radius of the $RD_{\alpha}$-matrix among all connected graphs of fixed order and precise vertex connectivity, edge connectivity, chromatic number and independence number, respectively., Comment: 21 pages