On the Spectrum of a Complete Multipartite Graph

Abstract

The spectrum S ( G ) of a graph G is defined as the sequence of eigenvalues of its adjacency matrix. The spectrum of a complete multipartite graph K has several remarkable properties. John Smith has shown that a graph has exactly one positive eigenvalue if and only if the non-isolated points form a complete multipartite graph. We now prove several additional properties of S ( K ). In the Interlacing Theorem for complete multipartite graphs K = K ( p 1 , p 2 , ..., p n ) where the parts p i are non-decreasing, we show that the n -1 negative eigenvalues in S ( K ) are respectively bounded by - p i and - p i +1 . We then find that no complete multipartite graph has a cospectral mate, amongst the complete multipartite graphs, a fact which enables us to establish a linear order, based on the value of λ 1 , among all K with p points and n parts. The minimal and maximal K in this ordering are seen to be respectively the complete multipartite graph in which all but one part is 1, and the graph in which the parts form an equipartition of p . We conclude with a criterion for K to have an integral spectrum.