On 3-isoregularity of multicirculants

TL;DR

This paper investigates the properties of 3-isoregular multicirculant graphs, proving non-existence results for certain classes and advancing understanding of their structure in relation to group theory.

Contribution

It establishes the non-existence of 3-isoregular bicirculants of order twice an odd number and provides partial results for other cases, linking graph regularity to group classification.

Findings

No 3-isoregular bicirculant of order twice an odd number exists.

Partial results for bicirculants of order twice an even number.

Connections to the classification of finite simple groups.

Abstract

A graph is said to be $k$-{\em isoregular} if any two vertex subsets of cardinality at most $k$, that induce subgraphs of the same isomorphism type, have the same number of neighbors. It is shown that no $3$-isoregular bicirculant (and more generally, no locally $3$-isoregular bicirculant) of order twice an odd number exists. Further, partial results for bicirculants of order twice an even number as well as tricirculants of specific orders, are also obtained. Since $3$-isoregular graphs are necessarily strongly regular, the above result about bicirculants, among other, brings us a step closer to obtaining a direct proof of a classical consequence of the Classification of Finite Simple Groups that no simply primitive group of degree twice a prime exists for primes greater than $5$.