Properties of the Extended Graph Permanent

TL;DR

The paper introduces the extended graph permanent, an infinite sequence invariant for all graphs, constructed via a novel polynomial and linked to point counts over finite fields, extending previous graph invariants.

Contribution

It develops the extended graph permanent as a new invariant for all graphs and connects it to a novel polynomial and finite field point counts.

Findings

Extended graph permanent is invariant under known period-preserving graph operations.

The sequence can be generated from a new graph polynomial via point counts over finite fields.

The construction generalizes the original graph permanent to all graphs.

Abstract

Previously, the graph permanent was introduced as a single-valued invariant for graphs $G$ with $|E(G)| = k(|V(G)|-1)$ for some $k \in \mathbb{Z}_{>0}$. Herein, we construct the extended graph permanent, an infinite sequence for all graphs. We prove that, like the graph permanent, the extended graph permanent is invariant under the graph operations that are known to preserve the period. Further, the original construction and extension arise from permanents of matrices, but we construct a novel graph polynomial such that the sequence can be generated from the point count of this polynomial, as a residue over prime-order finite fields.