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

TL;DR

This paper introduces an infinite family of highly symmetric 6-regular graphs derived from the Petersen graph, including new Ramanujan graphs, with detailed properties and computational reproducibility.

Contribution

It constructs an infinite family of 6-regular graphs from Petersen graphs, providing new Ramanujan examples and analyzing their symmetry and spectral properties.

Findings

G_3 and G_4 are Ramanujan graphs.

The graphs have automorphism group D_{5n}.

The construction yields highly symmetric regular graphs.

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.