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 \(\Phi\)-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.