On the characteristic polynomial of the adjacency matrix of the subdivision graph of a graph

The characteristic polynomial of the adjacency matrix of a graph is noted in connection with a quantity characterizing the topological nature of structural isomers saturated hydrocarbons [S], a set of numbers that are the same for all graphs isomorphic to the graph, and others [l]. Many properties of the characteristic polynomials of the adjacency matrices of a graph and its line graph [3] have so far been investigated by some researchers (see [ll, PI and PI)= In this note, the characteristic polynomial of the adjacency matrix of the subdivision graph [3] of a graph G is related to the characteristic polynomials of the adjacency matrices of G and its line graph. Let G be a graph with neither self-loops nor parallel edges, and let n and m be the numbers of vertices and edges of G, respectively. Let D be the incidence matrix whose ith row and jth column element Dij (i = 1,2, . . . , n ; i = 1,2, . . . , m) is 1 if the corresponding edge i is incident at the corresponding vertex i in G and 0 otherwise, and let D’ be the transpose of D. Moreover, let V be the diagonal matrix of order n whose ith diagonal element Vii is the number of all edges incident at the corresponding vertex i of G, and let