Cayley colour integral groups

TL;DR

This paper introduces and characterizes new classes of finite groups based on the integrality of their Cayley graphs with various coloring and normality conditions, expanding the understanding of Cayley integral groups.

Contribution

It defines three extensions of Cayley integral groups, characterizes two completely, and relates the third to inverse semi-rational groups, establishing an inclusion hierarchy.

Findings

Characterization of Cayley colour integral groups

Complete description of $rak{F}$-Cayley colour integral groups

Identification of normal Cayley integral groups with inverse semi-rational groups

Abstract

A finite group $G$ is said to be Cayley integral if every undirected Cayley graph $\operatorname{Cay}(G,S)$ on $G$ is integral. In this paper, we introduce three natural extensions of this concept; namely as: Cayley colour integral, $\mathfrak{F}$-Cayley colour integral and normal Cayley integral groups. We characterize the first two families in its entirety. The last family of groups is shown to be coinciding with inverse semi-rational groups introduced by Chillag and Dolfi, thereby providing an alternative characterization for the same. We also establish an inclusion hierarchy among these families.