An Infinite Family of 6_Regular B-Cayley Graphs from the Petersen Graph

Abstract

We construct an infinite family of 6-regular graphs $\{G_n{\}}_{n\ge 3}$ by taking $n$ copies of the Petersen graph and wiring corresponding vertices according to an $n$-cycle permutation. Each $G_n$ has $10n$ vertices, $30n$ edges, and automorphism group $D_{5n}$ of order $10n$, acting with two vertex orbits of size $5n$. The graphs have girth $4$ and diameter $\lfloor n/2\rfloor +2$. We prove that $G_3$ and $G_4$ are Ramanujan graphs, satisfying $|{\lambda }_2|\le 2\sqrt{5}$. The first five members ($n=3,\dots ,7$) have been deposited in the House of Graphs database as entries 56324--56328. This construction provides new examples of highly symmetric regular graphs and contributes two new Ramanujan graphs to the literature. All computational scripts are available online for full reproducibility.