On The Fixed Number of Graphs

TL;DR

This paper investigates the fixed number of graphs, introduces constructions for graphs with higher fixed numbers, and establishes bounds relating fixed number to diameter in distance-transitive graphs.

Contribution

It provides new insights into the fixed number of graphs, including methods to construct graphs with higher fixed numbers and bounds based on graph diameter.

Findings

Construction methods for graphs with higher fixed numbers

Bounds on fixed number in terms of diameter for distance-transitive graphs

Analysis of fixed and fixing numbers in graph automorphisms

Abstract

An automorphism on a graph $G$ is a bijective mapping on the vertex set $V(G)$, which preserves the relation of adjacency between any two vertices of $G$. An automorphism $g$ fixes a vertex $v$ if $g$ maps $v$ onto itself. The stabilizer of a set $S$ of vertices is the set of all automorphisms that fix vertices of $S$. A set $F$ is called fixing set of $G$, if its stabilizer is trivial. The fixing number of a graph is the cardinality of a smallest fixing set. The fixed number of a graph $G$ is the minimum $k$, such that every $k$-set of vertices of $G$ is a fixing set of $G$. A graph $G$ is called a $k$-fixed graph if its fixing number and fixed number are both $k$. In this paper, we study the fixed number of a graph and give construction of a graph of higher fixed number from graph with lower fixed number. We find bound on $k$ in terms of diameter $d$ of a distance-transitive $k$-fixed graph.