On the Spectrum of a Complete Multipartite Graph

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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 (p1, p2, ..., pn) where the parts pi are non-decreasing, we show that the n-1 negative eigenvalues in S(K) are respectively bounded by -pi and -pi+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 λ1, 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.

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Visiting Scholar, University of Michigan, 1978-79, Research supported by Deutsche Forschungsgemeinschaft, Bonn.