Enumeration of copermanental graphs

TL;DR

This paper computes the permanental polynomials for all graphs up to 11 vertices, analyzes their uniqueness, and suggests that the permanental polynomial may be more effective than the characteristic polynomial in graph characterization.

Contribution

It provides a comprehensive enumeration of permanental polynomials for small graphs and compares their effectiveness to characteristic polynomials in graph identification.

Findings

Most graphs have unique permanental polynomials as size increases.

The fraction of graphs with a copermanental mate tends to zero with more vertices.

Permanental polynomial outperforms characteristic polynomial in graph characterization.

Abstract

Let $G$ be a graph and $A$ the adjacency matrix of $G$. The permanental polynomial of $G$ is defined as $\mathrm{per}(xI-A)$. In this paper some of the results from a numerical study of the permanental polynomials of graphs are presented. We determine the permanental polynomials for all graphs on at most 11 vertices, and count the numbers for which there is at least one other graph with the same permanental polynomial. The data give some indication that the fraction of graphs with a copermanental mate tends to zero as the number of vertices tends to infinity, and show that the permanental polynomial does be better than characteristic polynomial when we use them to characterize graphs.