Equitable partitions of regular graphs, and perfect sets in normal Cayley graphs

TL;DR

This paper investigates conditions under which regular graphs and normal Cayley graphs admit equitable partitions and perfect sets, providing necessary criteria based on group characters and graph structure.

Contribution

It derives general necessary conditions for equitable partitions in regular graphs and applies these to characterize perfect sets in normal Cayley graphs using group irreducible characters.

Findings

Necessary conditions for equitable partitions in regular graphs.

Characterization of perfect sets in normal Cayley graphs.

Conditions expressed via irreducible group characters.

Abstract

An equitable partition of a graph $\Ga$ is a partition $\{V_1, \ldots, V_m\}$ of its vertex set such that for each pair $i, j$ all vertices in $V_i$ have the same number of neighbours in $V_j$. When $m=2$, $V_1$ is called an $(a, b)$-perfect set in $\Ga$, where $a$ is the number of neighbours in $V_1$ of each vertex in $V_1$, and $b$ is the number of neighbours in $V_1$ of each vertex in $V_2$. In this paper we first derive general necessary conditions for a regular graph to admit two equitable partitions. As a corollary we obtain necessary conditions for the existence of an $(a,b)$-perfect set in a regular graph in terms of an arbitrary equitable partition. With the help of these results we then obtain necessary conditions for the existence of an $(a,b)$-perfect set in a normal Cayley graph in terms of the irreducible characters of the underlying group.