The rank distribution of matrices representing graphs with a long induced path over the field of two elements

Abstract

A square matrix $M$ represents a graph $\Gamma$ if its nonzero off-diagonal entries encode the adjacencies of $\Gamma$, subject to a fixed ordering of the vertices. Over the field of two elements, we investigate the distribution of ranks in the affine space consisting of all matrices representing a given $\Gamma$. In particular, we consider which graphs of order $n$ are represented by more matrices of rank $n-1$ than of rank $n$. This property reflects an exceptional feature of the space $M_n(\mathbb{F}_2)$ of all $n\times n$ matrices over $\mathbb{F}_2$, namely that its most frequently occurring rank is not $n$ but $n-1$. Our analysis focuses on the class of connected graphs with an induced path on all but one vertex. The main result is a characterisation of all such graphs that are represented by more matrices of rank $n-1$ than of rank $n$ over $\mathbb{F}_2$.