On some automorphism related parameters in graphs

TL;DR

This paper explores properties of $f$-sets in connected graphs and introduces the fixing share and fixing percentage to quantify the contribution of vertices in fixing sets, advancing understanding of graph automorphisms.

Contribution

It introduces the concepts of fixing share and fixing percentage, providing new tools to analyze the role of vertices in automorphism fixing sets in graphs.

Findings

Properties of $f$-sets in connected graphs are derived.

The fixing share and fixing percentage are defined and analyzed.

These concepts help quantify the contribution of each vertex in fixing automorphisms.

Abstract

In this paper, we deduce some properties of $f$-sets of connected graphs. Also, we introduce the concept of fixing share of each vertex of a fixing set $D$ to see the participation of each vertex when fixing a connected graph $G$. We define a parameter, called the fixing percentage, by using the concept of fixing share, which is helpful in determining the measure of the amount of fixing done by the elements of $D$ in $G$.