  In 1960, Hoffman and Singleton \cite{HS60} solved a celebrated equation for
square matrices of order $n$, which can be written as $$ (\kappa - 1) I_n + J_n
- A A^{\rm T} = A$$ 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
$\kappa$, respectively. If $A$ is an incidence matrix of some configuration
$\cal C$ of type $n_\kappa$, then the left-hand side $\Theta(A):= (\kappa -
1)I_n + J_n - A A^{\rm T}$ is an adjacency matrix of the non--collinearity
graph $\Gamma$ of $\cal C$. In certain situations, $\Theta(A)$ is also an
incidence matrix of some $n_\kappa$ configuration, namely the neighbourhood
geometry of $\Gamma$ introduced by Lef\`evre-Percsy, Percsy, and Leemans
\cite{LPPL}.
  The matrix operator $\Theta$ can be reiterated and we pose the problem of
solving the generalised Hoffman--Singleton equation $\Theta^m(A)=A$. In
particular, we classify all $(0,1)$--matrices $M$ with all row and column sums
equal to $\kappa$, for $\kappa = 3,4$, 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}.
