Generalized Characteristic Polynomial of Generalized $R$-Vertex Corona

Abstract

Let $G$ be a graph of order $n$ with adjacency matrix $A(G)$ and diagonal degree matrix $D(G)$. The generalized characteristic polynomial of $G$ is defined to be $f_G(x,t)=det(x{I}_n–(A(G)–tD(G)))$. The $R$-graph of $G$, denoted by $R(G)$, is obtained by adding a new vertex for each edge of $G$ and joining each new vertex to both end vertices of the corresponding edge. The generalized $R$-vertex corona, denoted by $R(G)⊡{\wedge }_i^{n}H$, is the graph obtained from $R(G)$ and $H$ by joining the $i$-th vertex of $V(G)$ to every vertex of $H$. In this paper, we determine the generalized characteristic polynomial of $R(G)⊡{\wedge }_i^{n}H$. As applications, we get infinitely many pairs of generalized cospectral graphs, the number of spanning trees and Kirchhoff index of $R(G)⊡{\wedge }_i^{n}H$.