Hoffman-Singleton Graph Research Articles

Partially balanced incomplete block (PBIB)-designs arising from diametral paths in some strongly regular graphs

The partially balanced incomplete block (PBIB)-designs and association schemes arising from different graph parameters is a well-studied concept. In this paper, we define and construct PBIB-designs with association schemes in strongly regular graphs through the nodes belonging to the diametral paths as blocks. A [Formula: see text]-design, called a diametral-design (in short) over a strongly regular graph [Formula: see text] of degree [Formula: see text], diameter [Formula: see text], is an ordered pair [Formula: see text], where [Formula: see text] and [Formula: see text] is the set of all diametral paths of [Formula: see text], called blocks, containing the nodes belonging to the diametral paths, satisfying the following conditions: (i) If [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text] blocks containing [Formula: see text]; (ii) If [Formula: see text], [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text]-blocks containing [Formula: see text]. We construct PBIB-designs with two-association schemes through diametral paths corresponding to the strongly regular graphs such as the square lattice graphs [Formula: see text], the triangular graphs [Formula: see text], the three Chang graphs, the Petersen graph, the Shrikhande graph, the Clebsch graph, the complement of Clebsch graph, the complement of Schläfli graph, complete bipartite graphs, the Hoffman–Singleton graph, etc. and in each case we give the composition of the diametral paths. These serve as extensions for the collection of near impossible class of strongly regular graphs with given parameters having two-association schemes.

Majorana algebra for the Hoffman–Singleton graph

Majorana theory is an axiomatic tool for studying the Monster group M and its subgroups through the 196,884-dimensional Conway–Griess–Norton algebra. The theory was introduced by A. A. Ivanov in 2009 and since then it experienced a remarkable development including the classification of Majorana representations for small (and not so small) groups. The group U_3(5) is (isomorphic to) the socle of the centralizer in M of a subgroup of order 25. The involutions of this U_3(5)-subgroup are 2A-involutions in the Monster. Therefore, U_3(5) possesses a Majorana representation (based on the embedding in the Monster). We prove that this is the unique Majorana representation of U_3(5), calculate its dimension, which is 798, and obtain a description in terms of the Hoffman–Singleton graph of which the automorphism group has U_3(5) as an index 2 subgroup.

Factorizations of Almost Simple Groups with a Solvable Factor, and Cayley Graphs of Solvable Groups

A characterization is given for the factorizations of almost simple groups with a solvable factor. It turns out that there are only several infinite families of these nontrivial factorizations, and an almost simple group with such a factorization cannot have socle exceptional Lie type or orthogonal of minus type. The characterization is then applied to study s s -arc-transitive Cayley graphs of solvable groups, leading to a striking corollary that, except for cycles, a non-bipartite connected 3 3 -arc-transitive Cayley graph of a finite solvable group is necessarily a normal cover of the Petersen graph or the Hoffman-Singleton graph.

On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons

An edge coloring of a simple graph G is said to be proper rainbow-cycle-forbidding (PRCF, for short) if no two incident edges receive the same color and for any cycle in G, at least two edges of that cycle receive the same color. A graph G is defined to be PRCF-good if it admits a PRCF edge coloring, and G is deemed PRCF-bad otherwise. In recent work, Hoffman, et al. study PRCF edge colorings and find many examples of PRCF-bad graphs having girth less than or equal to 4. They then ask whether such graphs exist having girth greater than 4. In our work, we give a straightforward counting argument showing that the Hoffman-Singleton graph answers this question in the affirmative for the case of girth 5. It is then shown that certain generalized polygons, constructed of sufficiently large order, are also PRCF-bad, thus proving the existence of PRCF-bad graphs of girth 6, 8, 12, and 16.

A combinatorial basis for Terwilliger algebra modules of a bipartite distance-regular graph

Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and for any integer i, let Γi(x) denote the set of vertices at distance i from x. Let V=ℂX denote the vector space over ℂ consisting of column vectors whose coordinates are indexed by X and whose entries are in ℂ, and for z∈X let ẑ denote the element of V with a 1 in the z coordinate and 0 in all other coordinates. Fix vertices x,u,v where u∈Γ2(x) and v∈Γ2(x)∩Γ2(u), and let T=T(x) denote the Terwilliger algebra with respect to x. Under certain additional combinatorial assumptions, we give a combinatorially-defined spanning set for a T-module of endpoint 2,and we give the action of the adjacency matrix on this spanning set. The vectors in our spanning set are defined as sums and differences of vectors ẑ, where the vertices z are chosen based on the their distances from x,u, and v.We use this T-module to construct combinatorially-defined bases for all isomorphism classes of irreducible T-modules of endpoint 2 for examples including the Doubled Odd graphs, the Double Hoffman–Singleton graph, Tutte’s 12-cage graph, and the Foster graph. We provide a list of several other graphs satisfying our conditions.

Edge-transitive bi-Cayley graphs

A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. This generalisation of a Cayley graph gives a class of graphs that includes many important examples such as the Petersen graph, the Gray graph and the Hoffman-Singleton graph. A bi-Cayley graph Γ over a group H is called normal if H is normal in the full automorphism group of Γ, and almost-normal edge-transitive if the normaliser of H in the full automorphism group of Γ is transitive on the edges of Γ. In this paper, we give a characterisation of almost-normal edge-transitive bi-Cayley graphs, and in particular, we give a detailed description of 2-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic p-groups. We find that under certain conditions, ‘almost-normal edge-transitive’ is the same as ‘normal’ for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most 6. Also we answer some open questions from the literature about these and related classes of graphs, posed by Li (in Proc. Amer. Math. Soc. 133 (2005)), Marušič and Potočnik (in Eur. J. Comb. 22 (2001)) and Marušič and Šparl (in J. Algebr. Comb. 28 (2008)), and we correct a major error in a recent paper by Li, Song and Wang (in J. Comb. Theory Ser. A 120 (2013)).

On the quantum symmetry of distance-transitive graphs

In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs H(n,3), the Johnson graphs J(n,2) and the Kneser graphs K(n,2) do not have quantum symmetry. We also give a table with the quantum automorphism groups of all cubic distance-transitive graphs. Furthermore, with one graph missing, we can now decide whether or not a distance-regular graph of order ≤20 has quantum symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum symmetry. On a final note, we present an example of a pair of graphs with the same intersection array (the Shrikhande graph and the 4×4 rook's graph), where one of them has quantum symmetry and the other one does not.

New methods for finding minimum genus embeddings of graphs on orientable and non-orientable surfaces

The question of how to find the smallest genus of all embeddings of a given finite connected graph on an orientable (or non-orientable) surface has a long and interesting history. In this paper we introduce four new approaches to help answer this question, in both the orientable and non-orientable cases. One approach involves taking orbits of subgroups of the automorphism group on cycles of particular lengths in the graph as candidates for subsets of the faces of an embedding. Another uses properties of an auxiliary graph defined in terms of compatibility of these cycles. We also present two methods that make use of integer linear programming, to help determine bounds for the minimum genus, and to find minimum genus embeddings. This work was motivated by the problem of finding the minimum genus of the Hoffman-Singleton graph, and succeeded not only in solving that problem but also in answering several other open questions.

An algebraic approach to lifts of digraphs

We present some applications of a new matrix approach for studying the properties of the lift Γα of a voltage digraph, which has arcs weighted by the elements of a group. As a main result, when the involved group is Abelian, we completely determine the spectrum of Γα. As some examples of our technique, we study some basic properties of the Alegre digraph, and completely characterize the spectrum of a new family of digraphs, which contains the generalized Petersen graphs, and the Hoffman–Singleton graph.

Self-Dual Codes Related to the Rudvalis Group

In this note, we will show that there exist exactly three self-dual codes with the automorphism group isomorphic to the Rudvalis group up to equivalence. Furthermore we will give some descriptions of codewords which span these codes. Especially, we will describe some codewords by using a relation between the rank 3 graph of the Rudvalis group and the Hoffman–Singleton graph.

The Sylvester graph and Moore graphs

The combinatorial structure of a famous graph with large girth, namely the Sylvester graph, is studied. Simple techniques, such as two-way counting, partitions, circuit chasing and covers are used to identify smaller structures and to show that there are no other graphs that share a small number of regularity properties with it. As a consequence, we show the same for the Hoffman–Singleton graph. We notice that some much larger graphs with large girth have similar properties, and could be studied using the same techniques. In particular, we show that just as the Hoffman–Singleton graph contains the Sylvester graph, a Moore graph of valency 57, whose existence is a famous open problem, must contain a subgraph with a structure that is similar to the one we derived for the Sylvester graph.

A High-Performance and Cost-Efficient Interconnection Network for High-Density Servers

The flourishing large-scale and high-throughput web applications have emphasized the importance of high-density servers for their distinct advantages, such as high computing density, low power and low space requirements. To achieve above advantages, an efficient intra-server interconnection network is necessary. Most state-of-the-art high-density servers adopt the fully-connected intra-server network to achieve high network performance. Unfortunately, this solution is very expensive due to the high degree of nodes. To address this problem, we exploit the theory optimized moore graph to interconnect the chips within a server. Considering the size of applications, the 50-size moore graph, namely the Hoffman-Singleton graph, is extensively discussed in this paper. The simulation results show that it could attain comparative performance as the fully-connected network with much lower cost. In practice, however, chips could be integrated onto multiple boards. Thus, the graph should be divided into self-connected sub-graphs with the same size. Unfortunately, state-of-the-art solutions do not consider the production problem and generate heterogeneous sub graphs. To address this problem, we propose two equivalent-partition solutions for Hoffman-Singleton graph depending on the density of boards. Finally, we propose and evaluate a deadlock-free routing algorithm for each partition scheme.

Identifying Vertex Covers in Graphs

An identifying vertex cover in a graph $G$ is a subset $T$ of vertices in $G$ that has a nonempty intersection with every edge of $G$ such that $T$ distinguishes the edges, that is, $e \cap T \ne \emptyset$ for every edge $e$ in $G$ and $e \cap T \ne f \cap T$ for every two distinct edges $e$ and $f$ in $G$. The identifying vertex cover number $\tau_D(G)$ of $G$ is the minimum size of an identifying vertex cover in $G$. We observe that $\tau_D(G) + \rho(G) = |V(G)|$, where $\rho(G)$ denotes the packing number of $G$. We conjecture that if $G$ is a graph of order $n$ and size $m$ with maximum degree $\Delta$, then $\tau_D(G) \le \left( \frac{\Delta(\Delta - 1)}{\Delta^2 + 1} \right) n + \left( \frac{2}{\Delta^2 + 1} \right) m$. If the conjecture is true, then the bound is best possible for all $\Delta \ge 1$. We prove this conjecture when $\Delta \ge 1$ and $G$ is a $\Delta$-regular graph. The three known Moore graphs of diameter two, namely the $5$-cycle, the Petersen graph and the Hoffman-Singleton graph, are examples of regular graphs that achieves equality in the upper bound. We also prove this conjecture when $\Delta \in \{2,3\}$.

On terwilliger graphs in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph

The Hoffman-Singleton graph is the unique strongly regular graph with parameters (50, 7, 0, 1). A well-known hypothesis states that a distance-regular graph in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph has intersection array {50, 42, 1; 1, 2, 50} or {50, 42, 9; 1, 2, 42}. In the present paper, we prove this hypothesis under the assumption that a distance-regular graph is a Terwilliger graph and the graph diameter is at most 5.

Graphs in which neighborhoods of vertices are isomorphic to the Hoffman-Singleton graph

Connected graphs are studied in which neighborhoods of vertices are isomorphic to the Hoffman-Singleton graph (i.e., the strongly regular graph with parameters (50,7,0,1)). It is proved that a distance-regular graph in which neighborhoods of vertices are isomorphic to the Hoffman-Singleton graph has µ = 2.

On distance-regular graphs in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph

On distance-regular graphs in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph

On graphs in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph

On graphs in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph

A family of regular graphs of girth 5

Murty [A generalization of the Hoffman–Singleton graph, Ars Combin. 7 (1979) 191–193.] constructed a family of ( p m + 2 ) -regular graphs of girth five and order 2 p 2 m , where p ⩾ 5 is a prime, which includes the Hoffman–Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497–504]. This construction gives an upper bound for the least number f ( k ) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f ( k ) , k ⩾ 16 .

Some properties of the Hoffman-Singleton graph

The Hoffman-Singleton graph, with spectrum 7(1), 2(28), -3(21), is characterized among regular graphs by a star complement for the eigenvalue 2 that is, by an induced subgraph of order 22 without 2 as an eigenvalue. Properties of other induced subgraphs are noted; in particular, the subgraph induced by vertices at distance 2 from a given vertex is the edge-disjoint union of three Hamiltonian cycles.

Girth 5 graphs from relative difference sets

We consider the problem of construction of graphs with given degree k and girth 5 and as few vertices as possible. We give a construction of a family of girth 5 graphs based on relative difference sets. This family contains the smallest known graph of degree 8 and girth 5 which was constructed by Royle, four of the known cages including the Hoffman–Singleton graph, some graphs constructed by Exoo and some new smallest known graphs.