Spectra of subdivision-vertex join and subdivision-edge join of two graphs

The subdivision graph $\mathcal{S}(G)$ 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 \emph{subdivision-vertex join} of $G_1$ and $G_2$, denoted by $G_1\dot{\vee }{G}_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $V(G_1)$ with every vertex of $V(G_2)$. The \emph{subdivision-edge join} of $G_1$ and $G_2$, denoted by $G_1\underset{‾}{\vee }{G}_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $I(G_1)$ with every vertex of $V(G_2)$, where $I(G_1)$ is the set of inserted vertices of $\mathcal{S}(G_1)$. In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of $G_1\dot{\vee }{G}_2$ (respectively, $G_1\underset{‾}{\vee }{G}_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$, in terms of the corresponding spectra of $G_1$ and $G_2$. As applications, these results enable us to construct infinitely many pairs of cospectral graphs. We also give the number of the spanning trees and the Kirchhoff index of $G_1\dot{\vee }{G}_2$ (respectively, $G_1\underset{‾}{\vee }{G}_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$.