Automorphism groups and Distinguishing Colorings of Central and Middle Graphs

TL;DR

This paper investigates the automorphism groups of central and middle graphs derived from a base graph G, establishing their isomorphism for graphs of order at least 4 and applying findings to bounds on distinguishing parameters.

Contribution

It proves the isomorphism of automorphism groups of G, C(G), and M(G) for graphs with at least four vertices, and uses this to improve bounds on distinguishing numbers and indices.

Findings

Aut(G), Aut(C(G)), and Aut(M(G)) are isomorphic for graphs of order ≥ 4.

New upper bounds are derived for the distinguishing number and index of C(G) and M(G).

Results are inspired by an existing algorithm from 2016.

Abstract

Let G be a simple, finite, connected, and undirected graph. The middle graph M(G) of G is obtained from the subdivision graph S(G) after joining pairs of subdivided vertices that lie on adjacent edges of G and the central graph C(G) of G is obtained from S(G) after joining all non-adjacent vertices of G. We show that if the order of G is at least 4, then Aut(G), Aut(C(G)), and Aut(M(G)) are isomorphic (as abstract groups) and apply these results to obtain new upper bounds of the distinguishing number and the distinguishing index of C(G) and M(G) inspired by an algorithm due to Kalinowski, Pilsniak, and Wozniak from 2016.