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 $$(\hat{I}º-1)I_n+J_n-A{A}^{\mathrm{T}}=A$$ 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 $\hat{I}º$, respectively. If $A$ is an incidence matrix of some configuration $\mathcal{C}$ of type $n_{\hat{I}}º$, then the left-hand side $\hat{I}˜(A):=(\hat{I}º-1)I_n+J_n-A{A}^{\mathrm{T}}$ is an adjacency matrix of the non--collinearity graph $\hat{I}“$ of $\mathcal{C}$. In certain situations, $\hat{I}˜(A)$ is also an incidence matrix of some $n_{\hat{I}}º$ configuration, namely the neighbourhood geometry of $\hat{I}“$ introduced by Lefèvre-Percsy, Percsy, and Leemans \cite{LPPL}. The matrix operator $\hat{I}˜$ can be reiterated and we pose the problem of solving the generalised Hoffman--Singleton equation $\hat{I}{˜}^m(A)=A$. In particular, we classify all $(0,1)$--matrices $M$ with all row and column sums equal to $\hat{I}º$, for $\hat{I}º=3,4$, which are solutions of this equation. As a by--product, we obtain characterisations for incidence matrices of the configuration $10_{3}F$ in Kantor's list \cite{Kantor} and the $17_4$ configuration $#1971$ in Betten and Betten's list \cite{BB99}.