Degree distributions for a class of Circulant graphs

TL;DR

This paper characterizes when Cayley graphs are equivalent or weakly equivalent, computes degree distribution polynomials for various circulant graphs, and provides enumeration formulas for their weak equivalence classes.

Contribution

It introduces new characterizations of Cayley graph equivalence and derives degree distribution polynomials for specific classes of circulant graphs.

Findings

Degree distribution polynomials for circulant graphs of prime power order

Enumeration formulas for weak equivalence classes of these graphs

Characterizations of Cayley graph equivalence and weak equivalence

Abstract

We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group $\C{A}$. Using these characterizations, we find degree distribution polynomials for weak equivalence of some graphs including 1) circulant graphs of prime power order, 2) circulant graphs of order $4p$, 3) circulant graphs of square free order and 4) Cayley graphs of order $p$ or $2p$. As an application, we find an enumeration formula for the number of weak equivalence classes of circulant graphs of prime power order, order $4p$ and square free order and Cayley graphs of order $p$ or $2p$.