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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

We study the existence of quantum state transfer in $\mathcal{Q}$-graphs in this paper. The $\mathcal{Q}$-graph of a graph $G$, denoted by $\mathcal{Q}(G)$, is the graph derived from $G$ by plugging a new vertex to each edge of $G$ and joining two new vertices which lie on adjacent edges of $G$ by an edge. We show that, if all eigenvalues of a regular graph $G$ are integers, then its $\mathcal{Q}$-graph $\mathcal{Q}(G)$ has no perfect state transfer. In contrast, we also prove that the $\mathcal{Q}$-graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of $\mathcal{Q}$-graphs having no perfect state transfer, but admitting pretty good state transfer., Comment: 17 pages. arXiv admin note: text overlap with arXiv:2108.01325

The $\mathcal{Q}$-graph of a graph $G$, denoted by $\mathcal{Q}(G)$, is the graph derived from $G$ by plugging a new vertex to each edge of $G$ and adding a new edge between two new vertices which lie on adjacent edges of $G$. In this paper, we consider to study the existence of the signless Laplacian perfect state transfer and signless Laplacian pretty good state transfer in $\mathcal{Q}$-graphs of graphs. We show that, if all the signless Laplacian eigenvalues of a regular graph $G$ are integers, then the $\mathcal{Q}$-graph of $G$ has no signless Laplacian perfect state transfer. We also give a sufficient condition that the $\mathcal{Q}$-graph of a regular graph has signless Laplacian pretty good state transfer when $G$ has signless Laplacian perfect state transfer between two specific vertices., Comment: 17 pages

Very recently, the first geometric-arithmetic index $GA$ and arithmetic-geometric index $AG$ were introduced in mathematical chemistry. In the present paper, we first obtain some lower and upper bounds on $AG$ and characterize the extremal graphs. We also establish various relations between $AG$ and other topological indices, such as the first geometric-arithmetic index $GA$, atom-bond-connectivity index $ABC$, symmetric division deg index $SDD$, chromatic number $\chi$ and so on. Finally, we present some sufficient conditions of $GA(G)>GA(G-e)$ or $AG(G)>AG(G-e)$ for an edge $e$ of a graph $G$. In particular, for the first geometric-arithmetic index, we also give a refinement of Bollob\'{a}s-Erd\H{o}s-type theorem obtained in [3]., Comment: 24 pages, 4 figures, 32 conferences

The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G), \;\;0\le\alpha \le 1,$$ where $D(G)$ and $Tr(G)$ denote the distance matrix and diagonal matrix of the vertex transmissions of $G$, respectively. This paper characterizes the unique graph with minimum generalized distance spectral radius among the connected graphs with fixed chromatic number, which answers a question about the generalized distance spectral radius in spectral extremal theories. In addition, we also determine graphs with minimum generalized distance spectral radius among the $n$-vertex trees and unicyclic graphs, respectively. These results generalize some known results about distance spectral radius and distance signless Laplacian spectral radius of graphs., Comment: 13 pages

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. Applying the Cauchy-Schwarz inequality and the Ozeki inequality, along with its refined version, we obtain some improved bounds on $LEL$ and $IE$ of the $\mathcal {R}$-graph and $\mathcal{Q}$-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., Comment: 18 pages, 32 conference

For simple graphs $G$, $G_1$ and $G_2$, we denote their double corona based on $R$-graph by $G^{(R)}\otimes{\{G_1,G_2\}}$. This paper determines the normalized Laplacian spectrum of $G^{(R)}\otimes{\{G_1,G_2\}}$ in terms of these of $G$, $G_1$ and $G_2$ whenever $G$, $G_1$ and $G_2$ are regular. The obtained result reduces to the normalized Laplacian spectra of the $R$-vertex corona $G^{(R)}\odot{G_1}$ and $R$-edge corona $G^{(R)}\circleddash{G_2}$ by choosing $G_2$ or $G_1$ as a null-graph, respectively. Finally, applying the results of the paper, we construct infinitely many pairs of normalized Laplacian cospectral graphs., Comment: 9 pages, 19 conference

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 for the double join matrix, 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 the factor graphs. These results generalize some well-known results about some join operations of graphs., Comment: 15 pages, 2 figures, 21 conferences

For two simple graphs $G_1$ and $G_2$, we denote the subdivision-vertex join and subdivision-edge join of $G_1$ and $G_2$ by $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$, respectively. This paper determines the normalized Laplacian spectra of $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$ in terms of these of $G_1$ and $G_2$ whenever $G_1$ and $G_2$ are regular. As applications, we construct some non-regular normalized Laplacian cospectral graphs. Besides we also compute the number of spanning trees and the degree-Kirchhoff index of $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$ for regular graphs $G_1$ and $G_2$., Comment: 15 pages,19 confereces

Let $G[F,V_k,H_v]$ be the graph with $k$ pockets, where $F$ is a simple graph of order $n\geq1$, $V_k=\{v_1,\ldots,v_k\}$ is a subset of the vertex set of $F$ and $H_v$ is a simple graph of order $m\geq2$, $v$ 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 $n\geq2$, $E_k=\{e_1,\ldots,e_k\}$ is a subset of the edge set of $F$ and $H_{uv}$ is a simple graph of order $m\geq3$, $uv$ is a specified edge of $H_{uv}$ such that $H_{uv}-u$ is isomorphic to $H_{uv}-v$. 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 an application of these results, we construct infinitely many pairs of signless Laplacian (resp. adjacency) cospectral graphs., Comment: 16 pages, 0 figures

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 $W_Q(t)=\sum\nolimits_{k = 0}^\infty {N_k t^k }$. 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 $\overline{G}$. Using this result, we study some $Q$-spectral properties of graphs and compute the $Q$-polynomials for some graphs obtained by the use of some operation on graphs, such as the complement graph of a regular graph, the join of two graphs, the (edge)corona of two graphs and so forth. 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 $(\lambda I_n-Q(G))^{-1}$. 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 $Q$-coronals of the join of two graphs and the complete multipartite graphs.

In this paper, we give upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and characterize the equality cases. These bounds theoretically improve and generalize some known results of Duan et al.[X. Duan, B. Zhou, Sharp bounds on the spectral radius of a nonnegative matrix, Linear Algebra Appl. (2013), http://dx.doi.org/10.1016/j.laa.2013.08.026]. Finally, applying these bounds to various matrices associated with a graph, we obtain some new upper and lower bounds on various spectral radiuses of graphs, which generalize and improve some known results., Comment: 10 pages, 1 figures, 15 conference