On 3-isoregularity of multicirculants

TL;DR

The paper investigates 3-isoregularity within multicirculants, focusing on bicirculants and tricirculants with semiregular automorphisms. It develops a symbol-based framework for analyzing strongly regular multicirculants, derives explicit parameter constraints, and applies Hoffman-type bounds to rule out 3-isoregularity in many families. The main results show nonexistence of locally 3-isoregular SR $n$-bicirculants when $n$ is odd, and provide partial nonexistence results for even $n$ as well as for tricirculants, thereby advancing toward understanding the role of isoregularity in the broader context of finite permutation groups and CFSG-related questions. These findings connect combinatorial regularity conditions to group actions and offer concrete boundary cases (e.g., $n$-bicirculants with $n=2m^2$ and small $m$) that guide future classifications.

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 $5$.

Theorems & Definitions (24)

• Proposition 2.1
• Proposition 2.2
• Corollary 2.3
• Theorem 2.4
• Example 2.5
• Example 2.6
• Proposition 2.7
• Example 2.8
• Proposition 3.1
• Proposition 3.2