Higman-Sims Graph Research Articles

Note on the sum of the smallest and largest eigenvalues of a triangle-free graph

Let G be a triangle-free graph on n vertices with adjacency matrix eigenvalues μ1(G)≥μ2(G)≥…≥μn(G). In this paper we study the quantityμ1(G)+μn(G). We prove that for any triangle-free graph G we haveμ1(G)+μn(G)≤(3−22)n. This was proved for regular graphs by Brandt, we show that the condition on regularity is not necessary. We also prove that among triangle-free strongly regular graphs the Higman-Sims graph achieves the maximum ofμ1(G)+μn(G)n.

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 $50$.

Linear closures of finite geometries

The interrelations between finite geometries (finite incidence structures) and linear codes over finite fields are discussed under some special fundamental aspects. For any incidence structure \({\mathcal{I}}\) block codes, block-difference codes and co-block codes over finite fields of characteristic p are discussed resp. introduced; correspondingly p-modular co-blocks are defined for \({\mathcal{I}}\). Orthogonality modulo p is introduced as a concept relating different geometries having the same point set. Conversely three types of block-tactical geometries may be derived from vector classes of fixed Hamming weight in a given linear code. These geometries are tactical configurations if the given code admits a transitive permutation group. A combination of both approaches leads to the concept of p-closure of a finite geometry and to the notions of p-closed, weakly p-closed and p-dense incidence structures. These geometric concepts are applied to simple or directed graphs via their natural “adjacency geometry”. Here the above mentioned code theoretic treatment leads to the concept of p-modular co-adjacent vertex sets. As instructive examples the Petersen graph, its complemetary graph and the Higman-Sims graph are considered.

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 graphs in which neighborhoods of vertices are isomorphic to the Higman-Sims graph

The Higman-Sims graph is the unique strongly regular graph with parameters (100, 22, 0, 6). In this paper, amply regular graphs in which neighborhoods of vertices are isomorphic to the Higman-Sims graph are classified. This result continues the investigation of amply regular locally F-graphs, where F is the class of strongly regular graphs without triangles.

On graphs in which the neighborhood of each vertex is isomorphic to the Higman-Sims graph

On graphs in which the neighborhood of each vertex is isomorphic to the Higman-Sims graph

On finite edge-primitive and edge-quasiprimitive graphs

Many famous graphs are edge-primitive, for example, the Heawood graph, the Tutte–Coxeter graph and the Higman–Sims graph. In this paper we systematically analyse edge-primitive and edge-quasiprimitive graphs via the O'Nan–Scott Theorem to determine the possible edge and vertex actions of such graphs. Many interesting examples are given and we also determine all G-edge-primitive graphs for G an almost simple group with socle PSL ( 2 , q ) .

On automorphisms of 6-covers of the Higman-Sims graph

On automorphisms of 6-covers of the Higman-Sims graph

Covers of the Higman-Sims graph and their automorphisms

Covers of the Higman-Sims graph and their automorphisms

Automorphisms of Strongly Regular Krein Graphs without Triangles

A strongly regular graph is called a Krein graph if, in one of the Krein conditions, an equality obtains for it. A strongly regular Krein graph Kre(r) without triangles has parameters ((r2 + 3r)2, r3 + 3r2 + r, 0, r2 + r). It is known that Kre(1) is a Klebsh graph, Kre(2) is a Higman-Sims graph, and that a graph of type Kre(3) does not exist. Let G be the automorphism group of a hypothetical graph Γ = Kre(5), g be an element of odd prime order p in G, and Ω = Fix(g). It is proved that either Ω is the empty graph and p = 5, or Ω is a one-vertex graph and p = 41, or Ω is a 2-clique and p = 17, or Ω is the complete bipartite graph K8,8, from which the maximal matching is removed, and p = 3.

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 $S(3,6,22)$.

The local density of triangle-free graphs

How dense can every induced subgraph of ⌊ αn⌋ vertices (0 < α ⩽ 1) of a triangle-free graph of order n be? Tools will be developed to estimate the local density of graphs, based on the spectrum of the graph and on a fractional viewpoint. These tools are used to refute a conjecture of Erdős et al. about the local density of triangle-free graphs for a certain range of α, by estimating the local density of the Higman-Sims graph via its eigenvalues. Moreover, the local density will be related to a long-standing conjecture of Erdős, saying that every triangle-free graph can be made bipartite by the omission of at most n 2/25 edges. Finally, a conjecture about the spectrum of regular triangle-free graphs is raised, which can be seen as a common relaxation of the two previous questions.

Jaeger's Higman–Sims State Model and theB2Spider

6discovered a remarkable checkerboard state model based on the Higman–Sims graph that yields a value of the Kauffman polynomial, which is a quantum invariant of links. We present a simple argument that the state model has the desired properties using the combinatorialB2spider10.

Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs

Some binary linear codes of length 50 and 100 are constructed using the adjacency matrices of the Hoffman-Singleton graph and the Higman-Sims graph, Some of the codes are optimal or nearly optimal for the given length and dimension. The dual codes admit majority logic decoding.

Spin models for link polynomials, strongly regular graphs and Jaeger’s Higman-Sims model

We recall first some known facts on Jones and Kauffman polynomials for links, and on state models for link invariants. We give next an exposition of a recent spin model due to F. Jaeger and which involves Higman-Sims graph. The associated invariant assigns to an oriented link evaluation for a = -τ 5 and z = 1 of its Kauffman polynomial in Dubrovnik form, where τ denotes golden ratio. 1. Introduction. A knot is a simple closed curve in R 3 and a link is a finite union of disjoint knots. We denote by L a link L together with an orientation on each of its components. Two oriented links L, L' are isotopic, and we write L' « L, if there exists a family (Φt)o<t< of homeomorphism s of R3 such that map [0, 1] —• R3 sending t to φt(x) is continuous for each x e R3 and such that φo = id, φ\(L) = L 1, where last equation indicates that orientations correspond via φ. Links considered here are always assumed to be tame, namely isotopic to links made of smoothly embedded curves. A Ω-valued invariant for oriented links is a map L »-• I(L) which associates to each oriented link L in R3 an element I(L) of some ring Ω, for example C or a ring of Laurent polynomials, in such a way that I(Lf) = I(L) whenever Lf « L. Classically, one of most studied example of link invariant is Alexander-Conway polynomial Δ(L) e Z[t±ι] defined by J. W. Alexander in 1928 [Ale], with a normalization made precise by J. H. Conway in 1969 [Con]; notation (L rather than L) indicates that, at least for knots, Δ(L) does not depend on choice of an orientation on knot. The polynomial invariant L —• Δ(L) is well understood in terms of standard algebraic topology (homology of the infinite cyclic covering of complement of L in R 3) see e.g. [Rha], [Rol] or [BuZ]. The subject entered a new era in 1984 [Jol] with discovery of Jones polynomial V(L) e Z[t±ι/2]. This was starting point of several other invariants, including Kauffman polynomial

Strongly regular graphs and spin models for the Kauffman polynomial

We study spin models for invariants of links as defined by Jones [22]. We consider the two algebras generated by the weight matrices of such models under ordinary or Hadamard product and establish an isomorphism between them. When these algebras coincide they form the Bose-Mesner algebra of a formally self-dual association scheme. We study the special case of strongly regular graphs, which is associated to a particularly interesting link invariant, the Kauffman polynomial [27]. This leads to a classification of spin models for the Kauffman polynomial in terms of formally self-dual strongly regular graphs with strongly regular subconstituents [7]. In particular we obtain a new model based on the Higman-Sims graph [17].