Adjacency Matrices of Configuration Graphs

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}

Sur les semi-quadriques en tant qu'espaces de Shult projectifs

Nous montrons que des résultats de [3] conduisent à une caractérisation des semi-quadriques et nous améliorons celle-ci.Lefèvre-Percsy Christiane. Sur les semi-quadriques en tant qu'espaces de Shult projectifs. In: Bulletin de la Classe des sciences, tome 63, 1977. pp. 160-164

New geometries for finite groups and polytopes

We describe two methods to obtain new geometries from classes of geometries whose diagram satisfy given conditions. This gives rise to lots of new geometries for finite groups, and in particular for sporadic groups. It also produces new thin geometries related to polytopes.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Concerning a characterisation of Buekenhout-Metz unitals

In [7], for the casesq even andq=3, a characterisation of the Buekenhout-Metz unitals inPG(2,q 2) was given. We complete this characterisation by proving the result forq>3Catherine T. Quinn and Rey Cass