Uniqueness of graph square roots of girth six

TL;DR

This paper proves that for graphs with girth at least 6, having isomorphic squares implies the graphs are isomorphic, extending previous results on trees and graphs with larger girth.

Contribution

It establishes a new uniqueness result for graph square roots of girth six, extending prior work on trees and larger girth graphs.

Findings

Graphs of girth at least 6 with isomorphic squares are isomorphic.

Extends known results from trees and girth at least 7 graphs.

Provides insights on reconstructing graphs from higher powers.

Abstract

We prove that if two graphs of girth at least 6 have isomorphic squares, then the graphs themselves are isomorphic. This is the best possible extension of the results of Ross and Harary on trees and the results of Farzad et al. on graphs of girth at least 7. We also make a remark on reconstruction of graphs from their higher powers.