Computing the permanental polynomial of $4k$-intercyclic bipartite graphs

TL;DR

This paper introduces a combinatorial method to compute the permanental polynomial of $4k$-intercyclic bipartite graphs using a modified characteristic polynomial, offering an alternative to Pfaffian orientation techniques.

Contribution

It provides a new combinatorial formula for the permanental polynomial of a specific class of bipartite graphs, expanding the tools for graph polynomial computation.

Findings

Derived an expression for the permanental polynomial in terms of the modified characteristic polynomial.

Applied the method specifically to $4k$-intercyclic bipartite graphs.

Presented a purely combinatorial approach as an alternative to existing Pfaffian orientation methods.

Abstract

Let $G$ be a bipartite graph with adjacency matrix $A(G)$. The characteristic polynomial $\phi(G,x)=\det(xI-A(G))$ and the permanental polynomial $\pi(G,x) = \text{per}(xI-A(G))$ are both graph invariants used to distinguish graphs. For bipartite graphs, we define the modified characteristic polynomial, which is obtained by changing the signs of some of the coefficients of $\phi(G,x)$. For $4k$-intercyclic bipartite graphs, i.e., those for which the removal of any $4k$-cycle results in a $C_{4k}$-free graph, we provide an expression for $\pi(G,x)$ in terms of the modified characteristic polynomial of the graph and its subgraphs. Our approach is purely combinatorial in contrast to the Pfaffian orientation method found in the literature to compute the permanental polynomial.