Largest regular multigraphs with three distinct eigenvalues

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 $k e 2,3,7,57$, we show an upper bound $nleq k^2-k+1$, with equality if and only if there exists a finite projective plane of order $k-1$ that admits a polarity.