Distance-regular Graphs Research Articles

Completely regular codes with different parameters giving the same distance-regular coset graphs

We construct several classes of completely regular codes with different parameters, but identical intersection array. Given a prime power q and any two natural numbers a,b, we construct completely transitive codes over different fields with covering radius ρ=min{a,b} and identical intersection array, specifically, one code over Fqr for each divisor r of a or b. As a corollary, for any prime power q, we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters.

Distance regularity in buildings and structure constants in Hecke algebras

In this paper we define generalised spheres in buildings using the simplicial structure and Weyl distance in the building, and we derive an explicit formula for the cardinality of these spheres. We prove a generalised notion of distance regularity in buildings, and develop a combinatorial formula for the cardinalities of intersections of generalised spheres. Motivated by the classical study of algebras associated to distance regular graphs we investigate the algebras and modules of Hecke operators arising from our generalised distance regularity, and prove isomorphisms between these algebras and more well known parabolic Hecke algebras. We conclude with applications of our main results to non-negativity of structure constants in parabolic Hecke algebras, commutativity of algebras of Hecke operators, double coset combinatorics in groups with BN-pairs, and random walks on the simplices of buildings.

About non equivalent completely regular codes with identical intersection array

We obtain several classes of completely regular codes with different parameters, but identical intersection array. Given a prime power q and any two natural numbers a, b, we construct completely transitive codes over different fields with covering radius ρ=min⁡{a,b} and identical intersection array, specifically, we construct one code over Fqr for each divisor r of a or b. As a corollary, for any prime power q, we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters.

Max-cut and extendability of matchings in distance-regular graphs

A connected graph G of even order v is called t-extendable if it contains a perfect matching, t<v/2 and any matching of t edges is contained in some perfect matching. The extendability of G is the maximum t such that G is t-extendable. Since its introduction by Plummer in the 1980s, this combinatorial parameter has been studied for many classes of interesting graphs. In 2005, Brouwer and Haemers proved that every distance-regular graph of even order is 1-extendable and in 2014, Cioabă and Li showed that any connected strongly regular graph of even order is 3-extendable except for a small number of exceptions.In this paper, we extend and generalize these results. We prove that all distance-regular graphs with diameter D≥3 are 2-extendable and we also obtain several better lower bounds for the extendability of distance-regular graphs of valency k≥3 that depend on k, λ and μ, where λ is the number of common neighbors of any two adjacent vertices and μ is the number of common neighbors of any two vertices in distance two. In many situations, we show that the extendability of a distance-regular graph with valency k grows linearly in k. We conjecture that the extendability of a distance-regular graph of even order and valency k is at least ⌈k/2⌉−1 and we prove this fact for bipartite distance-regular graphs.In course of this investigation, we obtain some new bounds for the max-cut and the independence number of distance-regular graphs in terms of their size and odd girth and we prove that our inequalities are incomparable with known eigenvalue bounds for these combinatorial parameters.

An Application of Hoffman Graphs for Spectral Characterizations of Graphs

In this paper, we present the first application of Hoffman graphs for spectral characterizations of graphs. In particular, we show that the 2-clique extension of the $(t+1)\times (t+1)$-grid is determined by its spectrum when $t$ is large enough. This result will help to show that the Grassmann graph $J_2(2D,D)$ is determined by its intersection numbers as a distance regular graph, if $D$ is large enough.

On automorphisms of a distance-regular graph with intersection array $\{99,84,30;1,6,54\}$

Ранее было доказано, что дистанционно регулярный граф, в котором окрестности вершин сильно регулярны с параметрами $(99,14,1,2)$, имеет массив пересечений $\{99,84,1;1,14,99\}$, $\{99,84,1;1,12,99\}$ или $\{99,84,30;1,6,54\}$. В работе найдены возможные автоморфизмы графа с массивом пересечений $\{99,84,30;1,6,54\}$. В частности, доказано, что данный граф не является вершинно симметричным.

AUTOMORPHISMS OF DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {25; 16; 1; 1; 8; 25}

Makhnev and Samoilenko have found parameters of strongly regular graphs with no more than 1000 vertices, which may be neighborhoods of vertices in antipodal distance-regular graph of diameter 3 and with \(\lambda=\mu\). They proposed the program of investigation vertex-symmetric antipodal distance-regular graphs of diameter 3 with \(\lambda=\mu\), in which neighborhoods of vertices are strongly regular. In this paper we consider neighborhoods of vertices with parameters \((25,8,3,2)\).

Nonsymmetric Askey–Wilson polynomials and Q-polynomial distance-regular graphs

In his famous theorem (1982), Douglas Leonard characterized the q-Racah polynomials and their relatives in the Askey scheme from the duality property of Q-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the q-Racah polynomials in the above situation. Let Γ denote a Q-polynomial distance-regular graph that contains a Delsarte clique C. Assume that Γ has q-Racah type. Fix a vertex x∈C. We partition the vertex set of Γ according to the path-length distance to both x and C. The linear span of the characteristic vectors corresponding to the cells in this partition has an irreducible module structure for the universal double affine Hecke algebra Hˆq of type (C1∨,C1). From this module, we naturally obtain a finite sequence of orthogonal Laurent polynomials. We prove the orthogonality relations for these polynomials, using the Hˆq-module and the theory of Leonard systems. Changing Hˆq by Hˆq−1 we show how our Laurent polynomials are related to the nonsymmetric Askey–Wilson polynomials, and therefore how our Laurent polynomials can be viewed as nonsymmetric q-Racah polynomials.

On automorphisms of a distance-regular graph with intersection array {39, 36, 1; 1, 2, 39}

Possible prime-order automorphisms and their fixed-point subgraphs are found for a hypothetical distance-regular graph with intersection array {39, 36, 1; 1, 2, 39}. It is shownthat graphs with intersection arrays {15, 12, 1; 1, 2, 15}, {35, 32, 1; 1, 2, 35}, and {39, 36, 1; 1, 2, 39} are not vertex-symmetric.

On automorphisms of a distance-regular graph with intersection array {204, 175, 48, 1; 1, 12, 175, 204}

A distance-regular graph Γ with intersection array {204, 175, 48, 1; 1, 12, 175, 204} is an AT4-graph, and the antipodal quotient \(\overline \Gamma \) has parameters (800, 204, 28, 60). Automorphisms of these graphs are found. In particular, neither of the two graphs is arc-transitive.

The distance-regular graphs with valency [formula omitted], diameter [formula omitted] and [formula omitted

Let Γ be a distance-regular graph with valency k and diameter D, and let x be a vertex of Γ. We denote by ki(0≤i≤D) the number of vertices at distance i from x. In this paper, we try to quantify the difference between antipodal and non-antipodal distance-regular graphs. We will look at the sum kD−1+kD, and consider the situation where kD−1+kD≤2k. If Γ is an antipodal distance-regular graph, then kD−1+kD=kD(k+1). It follows that either kD=1 or the graph is non-antipodal. And for a non-antipodal distance-regular graph, it was known that kD(kD−1)≥k and kD−1≥k both hold. So, this paper concerns on obtaining more detailed information on the number of vertices for a non-antipodal distance-regular graph. We first concentrate on the case where the diameter equals three. In this case, the condition kD+kD−1≤2k is equivalent to the condition that the number of vertices is at most 3k+1. And we extend this result to all diameters. We note that although the result of the diameter 3 case is a corollary of the result of all diameters, the main difficulty is the diameter 3 case, and that the diameter 3 case confirms the following conjecture: there is no primitive distance-regular graph with diameter 3 having the M-property.

On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint two

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 let A denote the adjacency matrix of Γ. For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A, E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D, Ei⁎ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim Ei⁎W≤1 for 0≤i≤D. By the endpoint of W we mean min{i|Ei⁎W≠0}. For 0≤i≤D, let Γi(z) denote the set of vertices in X that are distance i from vertex z. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. In this paper we prove the following are equivalent: (i) Δ2>0 and for 2≤i≤D−2 there exist complex scalars αi,βi with the following property: for all x,y,z∈X such that ∂(x,y)=2, ∂(x,z)=i, ∂(y,z)=i we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|; (ii) For all x∈X there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra T(x) with endpoint two, and these modules are thin.

The Hilton–Milner theorem for the distance-regular graphs of bilinear forms

Let V be an (n+l)-dimensional vector space over the finite field Fq with l≥n>0, and W be a fixed l-dimensional subspace of V. Suppose F is a non-trivial intersecting family of n-dimensional subspaces U of V with U∩W=0. In this paper, we give the tight upper bound for the size of F, and describe the structure of F which reaches the upper bound.

On the Terwilliger algebra of bipartite distance-regular graphs with [formula omitted] and [formula omitted

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 let A denote the adjacency matrix of Γ. For x∈X and for 0≤i≤D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. It is known that Δ2=0 implies that D≤5 or c2∈{1,2}.For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A,E0∗,E1∗,…,ED∗, where for 0≤i≤D, Ei∗ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W we mean min{i|Ei∗W≠0}.We find the structure of irreducible T-modules of endpoint 2 for graphs Γ which have the property that for 2≤i≤D−1, there exist complex scalars αi, βi such that for all x,y,z∈X with ∂(x,y)=2,∂(x,z)=i,∂(y,z)=i, we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|, in case when Δ2=0 and c2=2. The case when Δ2=0 and c2=1 is already studied by MacLean et al. [15].We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2 and it is not thin. We give a basis for this T-module, and we give the action of A on this basis.

On the Metric Dimension of Imprimitive Distance-Regular Graphs

A resolving set for a graph $${\Gamma}$$ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of $${\Gamma}$$ is the smallest size of a resolving set for $${\Gamma}$$ . Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distance-regular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs. We also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible.

Endomorphisms of Twisted Grassmann Graphs

A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. An interesting problem in graph theory is to distinguish whether a graph is a core. The twisted Grassmann graphs, constructed by van Dam and Koolen in (Invent Math 162:189---193, 2005), are the first known family of non-vertex-transitive distance-regular graphs with unbounded diameter. In this paper, we show that every twisted Grassmann graph is a pseudo-core.

Arc-transitive antipodal distance-regular covers of complete graphs related to [formula omitted

In this paper, we classify antipodal distance-regular graphs of diameter three that admit an arc-transitive action of SU3(q). In particular, we find a new infinite family of distance-regular antipodal r-covers of a complete graph on q3+1 vertices, where q is odd and r is any divisor of q+1 such that (q+1)∕r is odd. Further, we find several new constructions of arc-transitive antipodal distance-regular graphs of diameter three in case λ=μ.

2-Walk-regular graphs with a small number of vertices compared to the valency

In 2013, it was shown that, for a given real number α>2, there are only finitely many distance-regular graphs Γ with valency k and diameter D≥3 having at most αk vertices, except for the following two cases: (i) D=3 and Γ is imprimitive; (ii)D=4 and Γ is antipodal and bipartite. In this paper, we will generalize this result to 2-walk-regular graphs. In this case, also incidence graphs of certain group divisible designs appear.

The uniqueness of a distance-regular graph with intersection array {32,27,8,1;1,4,27,32} and related results

It is known that, up to isomorphism, there is a unique distance-regular graph Delta with intersection array {32,27;1,12} [equivalently, Delta is the unique strongly regular graph with parameters (105, 32, 4, 12)]. Here we investigate the distance-regular antipodal covers of Delta . We show that, up to isomorphism, there is just one distance-regular antipodal triple cover of Delta (a graph hat{Delta } discovered by the author over 20 years ago), proving that there is a unique distance-regular graph with intersection array {32,27,8,1;1,4,27,32}. In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of Delta , and so no distance-regular graph with intersection array {32,27,6,1;1,6,27,32}. We also show there is no distance-regular antipodal 4-cover of Delta , and so no distance-regular graph with intersection array {32,27,9,1;1,3,27,32}, and that there is no distance-regular antipodal 6-cover of Delta that is a double cover of hat{Delta }.

Some spectral and quasi-spectral characterizations of distance-regular graphs

In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth.