Largest regular multigraphs with three distinct eigenvalues

Abstract

We deal with connected k-regular multigraphs of order n that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given k. For k=2,3,7, the Moore graphs are largest. For k≠2,3,7,57, we show an upper bound n≤k2−k+1, with equality if and only if there exists a finite projective plane of order k−1 that admits a polarity.