The Hoffman-Singleton Graph and its Automorphisms

On the Graphs of Hoffman-Singleton and Higman-Sims

We propose a new elementary definition of the Higman-Sims graph in which the 100 vertices are parametrised with Z4 × Z5 × Z5 and adjacencies are described by linear and quadratic equations. This definition extends Robertson’s pentagonpentagram definition of the Hoffman-Singleton graph and is obtained by studying maximum cocliques of the Hoffman-Singleton graph in Robertson’s parametrisation. Th...

OD-characterization of $S_4(4)$ and its group of automorphisms

Let $G$ be a finite group and $pi(G)$ be the set of all prime divisors of $|G|$. The prime graph of $G$ is a simple graph $Gamma(G)$ with vertex set $pi(G)$ and two distinct vertices $p$ and $q$ in $pi(G)$ are adjacent by an edge if an only if $G$ has an element of order $pq$. In this case, we write $psim q$. Let $|G= p_1^{alpha_1}cdot p_2^{alpha_2}cdots p_k^{alpha_k}$, where $p_1

A construction of the McLaughlin graph from the Hoffman-Singleton graph

In this paper, we give a new construction of the McLaughlin graph from the Hoffman-Singleton graph.

Hoffman-Singleton Graph

The graphs of Hoffman-Singleton, Higman-Sims, McLaughlin and the Hermite curve of degree 6 in characteristic 5

We construct the graphs of Hoffman-Singleton, Higman-Sims and McLaughlin from certain relations on the set of non-singular conics totally tangent to the Hermitian curve of degree 6 in characteristic 5. We then interpret this geometric construction in terms of the subgroup structure of the automorphism group of this Hermitian curve.