Concerning Some Properties of Signed Graphs Associated With Specific Graphs

TL;DR

This paper develops a method to classify signed graphs up to switching isomorphism using automorphism group actions, and applies it to specific graphs like K5 and GP(7; 2), providing new insights into their automorphism groups.

Contribution

It introduces a novel approach to count and classify switching non-isomorphic signed graphs based on automorphism group actions, with explicit classifications for K5 and GP(7; 2).

Findings

Classified all switching non-isomorphic signed graphs for K5 and GP(7; 2).

Established a method using automorphism groups for counting signed graphs.

Derived new results on automorphism groups of these signed graphs.

Abstract

Two signed graphs are called switching isomorphic if one of them is isomorphic to a switching equivalent of the other. To determine the number of switching non-isomorphic signed graphs on a specific graph, we will establish a method based on the action of its automorphism group. As an application and computational results, we classify all the switching non-isomorphic signed graphs arising from the complete graph K5 and the generalized Petersen graph GP(7; 2). Moreover, some results on the automorphism groups of the target signed graphs are obtained.