Lower Bounds for the Pfaffian Number of Graphs

TL;DR

This paper establishes the first known lower bounds for the pfaffian number of graphs, linking graph properties with matrix rank bounds and demonstrating that some graphs have arbitrarily large pfaffian numbers.

Contribution

It introduces the first lower bounds for the pfaffian number of graphs and relates these bounds to matrix rank properties, expanding understanding of graph Pfaffian characteristics.

Findings

Established lower bounds for the pfaffian number of graphs

Proved an upper bound for the rank of matrices related to the Khatri-Rao product

Showed existence of graphs with arbitrarily large pfaffian numbers

Abstract

The number of perfect matchings of a $k$-pfaffian graph can be counted by computing a linear combination of the pfaffians of $k$ matrices. The pfaffian number of a graph $G$ is the smallest integer $k$ such that $G$ is $k$-pfaffian. We present the first known lower bounds for the pfaffian number of graphs. As an intermediate step, we prove an upper bound for the rank of two matrices related to their Khatri-Rao product, a result of independent relevance. One of the consequences of these results is the existence of graphs whose pfaffian numbers are arbitrarily large.