# Largest regular multigraphs with three distinct eigenvalues

## TL;DR

This paper investigates the maximum size of connected regular multigraphs with exactly three distinct eigenvalues, establishing bounds and conditions for their existence based on algebraic and combinatorial properties.

## Contribution

It provides an upper bound on the number of vertices for such graphs and characterizes cases of equality linked to finite projective planes and polarities.

## Key findings

- For most degrees k, the maximum number of vertices is bounded by k^2 - k + 1.
- Moore graphs are the largest for k=2,3,7.
- Existence of certain graphs is tied to finite projective planes with polarities.

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