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 $|λ_2| \le 2\sqrt{5}$. 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.

Figures (2)

• Figure 1: The wiring pattern for $n=3$: each directed meta-edge creates a 3-cycle of connections. The three copies are shown as horizontal layers, with vertical edges representing the $n$-cycle wiring.
• Figure 2: A 2D force-directed canonical projection of $G_3$ generated via spring layout, illustrating the dense 6-regular structure.

Theorems & Definitions (24)

• Definition 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 4.1
• proof
• Corollary 4.2