Numerical Measures for Two-Graphs

TL;DR

This paper introduces numerical measures to distinguish two-graphs, proposes conjectures about finite differentiating conditions, and verifies these conjectures for specific regular two-graphs on 26 vertices.

Contribution

It presents new numerical measures for two-graphs and formulates conjectures linking these measures to graph differentiation, supported by verification on particular cases.

Findings

Conjectures hold for certain regular two-graphs on 26 vertices.

Numerical measures can distinguish two-graphs under specific conditions.

Verification supports the proposed finite differentiating criteria.

Abstract

We study characteristics which might distinguish two-graphs by introducing different numerical measures on the collection of graphs on $n$ vertices. Two conjectures are stated, one using these numerical measures and the other using the deck of a graph, which suggest that there is a finite set of conditions differentiating two-graphs. We verify that, among the four non-trivial non-isomorphic regular two-graphs on 26 vertices, both conjectures hold.