Ramanujan graphs with diameter at most three

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.