The characteristic polynomial of a graph operation and its application

Let R(G) be the graph obtained from G by adding a new vertex corresponding to each edge of G and by joining each new vertex to the end vertices of the corresponding edge. Let RT(G) be the graph obtained from R(G) by adding a new edge corresponding to every vertex of G, and by joining each new edge to every vertex of G. In this paper, we deduce the explicit expressions of characteristic polynomial of RT(G) of a regular graph G. Based on which, we obtain the adjacency spectrum of RT(G). As an application of the obtained results, the adjacency cospectrum graphs can be constructed by the proposed graph operation approach.