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

A square matrix M represents a graph $Γ$ if its nonzero off-diagonal entries encode the adjacencies of $Γ$, 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 $Γ$. 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 Mₙ(F₂) of all n× n matrices over F₂, 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 F₂.