Some Minimal Graphs by Interlacing Eigenvalues

Let G be a simple graph with p vertices and let A(G) be the adjacency matrix of G. The characteristic polynomial of G is the characteristic polynomial of A(G) and roots of the characteristic equation are the eigenvalues of G. In this paper we compute the characteristic polynomial of a class of graphs and show that the same class of graphs are minimal. (Here minimal indicates graphs with diameter d and exactly d + 1 different eigenvalues).