On the Eigenvalues of Some Signed Graphs

Let G be a simple graph and A(G) be the adjacency matrix of G. The matrix S(G) = J - I 2A(G) is called the Seidel matrix of G, where I is an identity matrix and J is a square matrix all of whose entries are equal to 1. Clearly, if G is a graph of order n with no isolated vertex, then the Seidel matrix of G is also the adjacency matrix of a signed complete graph Kn whose negative edges induce G. In this paper, we study the Seidel eigenvalues of the complete multipartite graph K-n1;...; nk and investigate its Seidel characteristic polynomial. We show that if there are at least three parts of size n(i), for some i = 1;...; k, then K-n1;...; (nk) is determined, up to switching, by its Seidel spectrum.