Self-converse mixed graphs are extremely rare

TL;DR

This paper investigates the rarity of self-converse mixed graphs, providing theoretical evidence that such graphs are extremely uncommon as their proportion approaches zero.

Contribution

It offers a theoretical analysis demonstrating that the fraction of self-converse mixed graphs tends to zero, highlighting their scarcity in the graph space.

Findings

Self-converse mixed graphs are extremely rare.

The fraction of such graphs tends to zero.

Theoretical evidence supports their scarcity.

Abstract

A mixed graph is cospectral to its converse, with respect to the usual adjacency matrices. Hence, it is easy to see that a mixed graph whose eigenvalues occur uniquely, up to isomorphism, must be isomorphic to its converse. It is therefore natural to ask whether or not this is a common phenomenon. This note contains the theoretical evidence to confirm that the fraction of self-converse mixed graphs tends to zero.