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Cayley colour integral groups

Sauvik Poddar, Angsuman Das

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.

Cayley colour integral groups

Abstract

A finite group is said to be Cayley integral if every undirected Cayley graph on is integral. In this paper, we introduce three natural extensions of this concept; namely as: Cayley colour integral, -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.
Paper Structure (6 sections, 25 equations, 2 figures)

This paper contains 6 sections, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Preliminaries and notations
  4. Cayley colour integral (CCI) groups
  5. $\mathfrak{F}$-Cayley colour integral ($\mathfrak{F}$-$\mathfrak{C}$CI) groups
  6. Normal Cayley integral (NCI) groups

Figures (2)

  • Figure 1: Implications among the groups mentioned in Definition \ref{['definition-Cay-col-int-groups']}.
  • Figure 2: Pictorial representation of the hierarchy between the groups.