Ramanujan graphs with diameter at most three

TL;DR

This paper constructs a new class of Ramanujan graphs with diameter at most three by applying a specific graph transformation involving line graphs and complements to any connected regular graph with at least five vertices.

Contribution

It introduces a novel method to generate Ramanujan graphs with small diameter from arbitrary regular graphs, expanding the known constructions of such optimal expanders.

Findings

The graph al{R}(G) is Ramanujan for all regular graphs G with at least 5 vertices.

The diameter of al{R}(G) is at most three.

The construction applies to any connected regular graph, broadening the class of known Ramanujan graphs.

Abstract

For a simple graph $G$, the complement and the line graph of $G$ are denoted by $G^c$ and $L(G)$, respectively. In this paper, we show that for every simple connected regular graph $G$ with at least $5$ vertices, the graph $\mathcal{R}(G):=L(L(G)^c)^c$ is a Ramanujan graph with diameter at most three.