Adjacency Matrices of Configuration Graphs

Abstract

In 1960, Hoffman and Singleton \cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$ (κ- 1) I_n + J_n - A A^{\rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all one matrix, and a $(0,1)$--matrix with all row and column sums equal to $κ$, respectively. If $A$ is an incidence matrix of some configuration $\cal C$ of type $n_κ$, then the left-hand side $Θ(A):= (κ- 1)I_n + J_n - A A^{\rm T}$ is an adjacency matrix of the non--collinearity graph $Γ$ of $\cal C$. In certain situations, $Θ(A)$ is also an incidence matrix of some $n_κ$ configuration, namely the neighbourhood geometry of $Γ$ introduced by Lefèvre-Percsy, Percsy, and Leemans \cite{LPPL}. The matrix operator $Θ$ can be reiterated and we pose the problem of solving the generalised Hoffman--Singleton equation $Θ^m(A)=A$. In particular, we classify all $(0,1)$--matrices $M$ with all row and column sums equal to $κ$, for $κ= 3,4$, which are solutions of this equation. As a by--product, we obtain characterisations for incidence matrices of the configuration $10_3F$ in Kantor's list \cite{Kantor} and the $17_4$ configuration $#1971$ in Betten and Betten's list \cite{BB99}.