On non-cyclic graph of finite groups

TL;DR

This paper investigates algebraic properties of non-cyclic graphs of finite groups, characterizing specific group structures based on graph isomorphisms and domination number sums.

Contribution

It provides a characterization of groups with particular non-cyclic graph structures and sums of domination numbers, specifically identifying groups isomorphic to D8 and D10.

Findings

Non-cyclic graph of D8 is isomorphic to K3 union isolated vertices.

Non-cyclic graph of D10 is isomorphic to K4 union isolated vertices.

Characterization of groups with domination number sums in {n, n-1, n-2, n-3}.

Abstract

Here we study some algebraic properties of non-cyclic graphs. In this paper we show that $\overline{\Gamma}_G$ is isomorphic to $K_3\cup (n-4)K_1$ or $K_4\cup (n-5)K_1$ if and only if $G$ is isomorphic to $D_8$ or $D_{10}$, respectively. We characterize all groups of order $n$ like $G$ in which $\gamma(\Gamma_G)+\gamma(\overline{\Gamma}_G)\in \{n, n-1, n-2, n-3\}$, where $|Cyc(G)|=1$.