On the Multiplicity of-1 and 1 in Signed Complete Graphs

Let Gamma = (G, sigma) be a signed graph, where G is the underlying simple graph and sigma : E(G) -> {-, +} is the sign function on the edges of G. The adjacency matrix of a signed graph has -1 or +1 for adjacent vertices, depending on the sign of the connecting edges. In this paper, we study the multiplicity of eigenvalues -1 and 1 for the signed complete graphs. Also, we determine the characteristic polynomial of a signed complete graph whose negative edges induce a complete tripartite graph.