The spectral characterizations of the connected multicone graphs Kw ▽ LHS and Kw ▽ LGQ(3,9)

In the past decades, graphs that are determined by their spectrum have received much more and more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. The main aim of this study is to characterize two classes of multicone graphs which are determined by their adjacency, Laplacian and signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let [Formula: see text] denote a complete graph on [Formula: see text] vertices. In the paper, we show that multicone graphs [Formula: see text] and [Formula: see text] are determined by both their adjacency spectra and their Laplacian spectra, where [Formula: see text] and [Formula: see text] denote the Local Higman–Sims graph and the Local [Formula: see text] graph, respectively. In addition, we prove that these multicone graphs are determined by their signless Laplacian spectra.

Guessing Games on Triangle-Free Graphs

The guessing game introduced by Riis [Electron. J. Combin. 2007] is a variant of the "guessing your own hats" game and can be played on any simple directed graph $G$ on $n$ vertices. For each digraph $G$, it is proved that there exists a unique guessing number $\mathrm{gn}(G)$ associated to the guessing game played on $G$. When we consider the directed edge to be bidirected, in other words, the graph $G$ is undirected, Christofides and Markström [Electron. J. Combin. 2011] introduced a method to bound the value of the guessing number from below using the fractional clique cover number $\kappa_f(G)$. In particular they showed $\mathrm{gn}(G) \geq |V(G)| - \kappa_f(G)$. Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph $G$ falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman-Sims graph has guessing number at least $77$ and at most $78$, while the bound given by fractional clique cover is $50$.

On the Codes Related to the Higman-Sims Graph

All linear codes of length $100$ over a field $F$ which admit the Higman-Sims simple group HS in its rank $3$ representation are determined. By group representation theory it is proved that they can all be understood as submodules of the permutation module $F\Omega$ where $\Omega$ denotes the vertex set of the Higman-Sims graph. This module is semisimple if $\mathrm{char} F\neq 2,5$ and absolutely indecomposable otherwise. Also if $\mathrm{char} F \in \{2, 5\}$ the submodule lattice is determined explicitly. The binary case $F = \mathbb{F}_2$ is studied in detail under coding theoretic aspects. The HS-orbits in the subcodes of dimension $\leq 23$ are computed explicitly and so also the weight enumerators are obtained. The weight enumerators of the dual codes are determined by MacWilliams transformation. Two fundamental methods are used: Let $v$ be the endomorphism determined by an adjacency matrix. Then in $H_{22} = \mathrm{Im} v $ the HS-orbits are determined as $v$-images of certain low weight vectors in $F\Omega$ which carry some special graph configurations. The second method consists in using the fact that $H_{23}/H_{21}$ is a Klein four group under addition, if $H_{23}$ denotes the code generated by $H_{22}$ and a "Higman vector" $x(m)$ of weight 50 associated to a heptad $m$ in the shortened Golay code $G_{22}$, and $H_{21}$ denotes the doubly even subcode of $H_{22}\leq H_{78} = {H_{22}}^\perp$. Using the mentioned observation about $H_{23}/H_{21}$ and the results on the HS-orbits in $H_{23}$ a model of G. Higman's geometry is constructed, which leads to a direct geometric proof that G. Higman's simple group is isomorphic to HS. Finally, it is shown that almost all maximal subgroups of the Higman-Sims group can be understood as stabilizers in HS of codewords in $H_{23}$.

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 ${\Bbb Z}_4\times{\Bbb Z}_5\times{\Bbb Z}_5$ and adjacencies are described by linear and quadratic equations. This definition extends Robertson's pentagon-pentagram definition of the Hoffman-Singleton graph and is obtained by studying maximum cocliques of the Hoffman-Singleton graph in Robertson's parametrisation. The new description is used to count the 704 Hoffman-Singleton subgraphs in the Higman-Sims graph, and to describe the two orbits of the simple group HS on them, including a description of the doubly transitive action of HS within the Higman-Sims graph. Numerous geometric connections are pointed out. As a by-product we also have a new construction of the Steiner system $S(3,6,22)$.