Computing the permanental polynomials of bipartite graphs by Pfaffian orientation

TL;DR

This paper investigates the conditions under which the permanental polynomial of bipartite graphs can be computed using Pfaffian orientations, establishing a link with planarity and cycle resonance, and provides methods for calculating these polynomials.

Contribution

It proves that the equality between permanental and characteristic polynomials holds only for bipartite graphs without even subdivisions of K_{2,3}, and characterizes such graphs as planar 1-cycle resonant.

Findings

The equality holds only if the bipartite graph contains no even subdivision of K_{2,3}.

Such graphs are proven to be planar.

A characterization of 2-connected bipartite graphs with no even subdivision of K_{2,3} as planar 1-cycle resonant.

Abstract

The permanental polynomial of a graph $G$ is $\pi(G,x)\triangleq\mathrm{per}(xI-A(G))$. From the result that a bipartite graph $G$ admits an orientation $G^e$ such that every cycle is oddly oriented if and only if it contains no even subdivision of $K_{2,3}$, Yan and Zhang showed that the permanental polynomial of such a bipartite graph $G$ can be expressed as the characteristic polynomial of the skew adjacency matrix $A(G^e)$. In this paper we first prove that this equality holds only if the bipartite graph $G$ contains no even subdivision of $K_{2,3}$. Then we prove that such bipartite graphs are planar. Further we mainly show that a 2-connected bipartite graph contains no even subdivision of $K_{2,3}$ if and only if it is planar 1-cycle resonant. This implies that each cycle is oddly oriented in any Pfaffian orientation of a 2-connected bipartite graph containing no even subdivision of $K_{2,3}$. As applications, permanental polynomials for some types of bipartite graphs are computed.