The Generalized Characteristic Polynomial of the Subdivision-Vertex and Subdivision-Edge Coronae

Abstract

The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex-disjoint graphs. The subdivision-vertex corona of $G_1$ and $G_2$, denoted by $G_1\odot G_2$, is the graph obtained from $S(G_1)$ and $|V(G_1)|$ copies of $G_2$, all vertex-disjoint, by joining the $i$th vertex of $V(G_1)$ to every vertex in the $i$th copy of $G_2$. The subdivision-edge corona of $G_1$ and $G_2$, denoted by $G_1\ominus G_2$, is the graph obtained from $S(G_1)$ and $|I(G_1)|$ copies of $G_2$, all vertex-disjoint, by joining the $i$th vertex of $I(G_1)$ to every vertex in the $i$th copy of $G_2$, where $I(G_1)$ is the set of inserted vertices of $S(G_1)$. In this paper, we determine the generalized characteristic polynomial of $G_1\odot G_2$ (respectively, $G_1\ominus G_2$). As applications, the results on the spectra of $G_1\odot G_2$ (respectively, $G_1\ominus G_2$) enable us to construct infinitely many pairs of $\mathrm{\Phi }$-cospectral graphs. The adjacency spectra of $G_1\odot G_2$ (respectively, $G_1\ominus G_2$) help us to construct many infinite families of integral graphs. By using the Laplacian spectra, we also obtain the number of spanning trees and Kirchhoff index of $G_1\odot G_2$ and $G_1\ominus G_2$, respectively.