The Spectrum of the Line Graph of the Subdivision Graph of the Complete Graph

A graph G is called integral if all eigenvalues of the adjacency matrix A(G) of G are integers. Let L2(Kp) = L(S(KP)) be the line graph of the subdivision graph S(KP) of the complete graph Kp. In this paper, we shall give the spectra and characteristic polynomials of S(KP) and L2(KP) from the theory on graphs. For the graph L2(KP), we derive the characteristic polynomials for its complement graph, its line graph, the complement graph of its line graph and the line graph of its complement graph. We also prove these graphs are integral graphs. The discovery of these integral graphs is a new contribution to the research of integral graphs.