Antipodal Distance-regular Graphs Research Articles

A New Feasibility Condition for the AT4 Family

Let $\Gamma$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $\theta_0>\theta_1>\theta_2>\theta_3>\theta_4$. Then $\Gamma$ is tight in the sense of Jurišić, Koolen, and Terwilliger (J. Algebraic Combin, 2000) whenever $\Gamma$ is locally strongly regular with nontrivial eigenvalues $p:=\theta_2$ and $-q:=\theta_3$. Assume that $\Gamma$ is tight. Then the intersection numbers of $\Gamma$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $\Gamma$. We denote $\Gamma$ by $\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph.Using this condition, we prove that there does not exist $\mathrm{AT4}(q^3-2q,q,2)$ for $q\equiv3$ $(\mathrm{mod}~4)$. We discuss the $\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$.

ON SOME VERTEX-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

In the present paper, we classify abelian antipodal distance-regular graphs \(\Gamma\) of diameter 3 with the following property: \((*)\) \(\Gamma\) has a transitive group of automorphisms \(\widetilde{G}\) that induces a primitive almost simple permutation group \(\widetilde{G}^{\Sigma}\) on the set \({\Sigma}\) of its antipodal classes. There are several infinite families of (arc-transitive) examples in the case when the permutation rank \({\rm rk}(\widetilde{G}^{\Sigma})\) of \(\widetilde{G}^{\Sigma}\) equals 2 moreover, all such graphs are now known. Here we focus on the case \({\rm rk}(\widetilde{G}^{\Sigma})=3\).Under this condition the socle of \(\widetilde{G}^{\Sigma}\) turns out to be either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, it was shown that the family of non-bipartite graphs \(\Gamma\) with the property \((*)\) such that \(rk(\widetilde{G}^{\Sigma})=3\) and the socle of \(\widetilde{G}^{\Sigma}\) is a sporadic or an alternating group is finite and limited to a small number of potential examples. The present paper is aimed to study the case of classical simple socle for \(\widetilde{G}^{\Sigma}\). We follow a classification scheme that is based on a reduction to minimal quotients of \(\Gamma\) that inherit the property \((*)\). For each given group \(\widetilde{G}^{\Sigma}\) with simple classical socle of degree \(|{\Sigma}|\le 2500\), we determine potential minimal quotients of \(\Gamma\), applying some previously developed techniques for bounding their spectrum and parameters in combination with the classification of primitive rank 3 groups of the corresponding type and associated rank 3 graphs. This allows us to essentially restrict the sets of feasible parameters of \(\Gamma\) in the case of classical socle for \(\widetilde{G}^{\Sigma}\) under condition \(|{\Sigma}|\le 2500.\)

Arc-transitive groups of automorphisms of antipodal distance-regular graphs of diameter 3 in affine case

Arc-transitive groups of automorphisms of antipodal distance-regular graphs of diameter 3 in affine case

Geometric Antipodal Distance-Regular Graphs with a Given Smallest Eigenvalue

In this paper we show that for a given integer $$m\ge 3$$, there are only finitely many antipodal distance-regular graphs with $$c_2\ge 2$$, odd diameter at least 5 and smallest eigenvalue at least $$-m$$. To prove this we first show that for a given integer $$m\ge 3$$, there are only finitely many antipodal distance-regular graphs with $$c_2\ge 2$$, diameter at least 5 and smallest eigenvalue at least $$-m$$ such that folded graphs are non-geometric. (A non-complete distance-regular graph is called geometric if it is the point graph of a partial linear space in which the set of lines is a set of Delsarte cliques.). Moreover, we show tight bounds for the diameter of geometric antipodal distance-regular graphs with an induced subgraph $$K_{2,1,1}$$ by studying parameters for geometric antipodal distance-regular graphs.

On the metric dimension of incidence graphs

A resolving set for a graph Γ 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 dimensionμ(Γ) is the smallest size of a resolving set for Γ. We consider the metric dimension of two families of incidence graphs: incidence graphs of symmetric designs, and incidence graphs of symmetric transversal designs (i.e. symmetric nets). These graphs are the bipartite distance-regular graphs of diameter 3, and the bipartite, antipodal distance-regular graphs of diameter 4, respectively. In each case, we use the probabilistic method in the manner used by Babai to obtain bounds on the metric dimension of strongly regular graphs, and are able to show that μ(Γ)=O(nlogn) (where n is the number of vertices).

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 \((25,8,3,2)\).

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.

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.

The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues

Let $\Gamma$ be a distance-regular graph with diameter $d$ and Kneser graph $K=\Gamma_d$, the distance-$d$ graph of $\Gamma$. We say that $\Gamma$ is partially antipodal when $K$ has fewer distinct eigenvalues than $\Gamma$. In particular, this is the case of antipodal distance-regular graphs ($K$ with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs ($K$ with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with $d$ distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance $d$ from every vertex. This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.

Equiangular lines and covers of the complete graph

The relation between equiangular sets of lines in the real space and distance-regular double covers of the complete graph is well known and studied since the work of Seidel and others in the 70s. The main topic of this paper is to continue the study on how complex equiangular lines relate to distance-regular covers of the complete graph with larger index. Given a set of equiangular lines meeting the relative (or Welch) bound, we show that if the entries of the corresponding Gram matrix are prime roots of unity, then these lines can be used to construct an antipodal distance-regular graph of diameter three. We also study in detail how the absolute (or Gerzon) bound for a set of equiangular lines can be used to derive bounds of the parameters of abelian distance-regular covers of the complete graph.

The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues

Let Γ be a distance-regular graph with diameter d and Kneser graph K=Γd, the distance-d graph of Γ. We say that Γ is partially antipodal when K has fewer distinct eigenvalues than Γ. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with d distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.

Two new infinite families of arc-transitive antipodal distance-regular graphs of diameter three with $$\lambda =\mu $$ λ = μ related to groups $$Sz(q)$$ S z ( q ) and $$^2G_2(q)$$ 2 G 2 ( q )

In this paper, we construct two new infinite families of arc-transitive distance-regular graphs, related to Suzuki groups $$Sz(q)$$Sz(q) and Ree groups $$^2G_2(q)$$2G2(q), where $$q>3$$q>3. They are antipodal $$r$$r-covers of complete graphs on $$q^2+1$$q2+1 or $$q^3+1$$q3+1 vertices, respectively, with $$\lambda =\mu $$?=μ and $$r>1$$r>1 being an arbitrary odd divisor of $$q-1$$q-1. We also find that the graph on the set of involutions of $$Sz(q)$$Sz(q) with $$q>3$$q>3, whose edges are the pairs of involutions $$\{u,v\}$${u,v} such that $$|uv|=5$$|uv|=5, is distance-regular.

Arc-transitive distance-regular covers of cliques with λ = µ

We study antipodal distance-regular graphs of diameter 3 such that their automorphism group acts transitively on the set of pairs (a, b), where {a, b} is an edge of the graph. Since the automorphism group of such graphs acts 2-transitively on the set of antipodal classes, the classification of 2-transitive permutation groups can be used. We classify arc-transitive distance-regular graphs of diameter 3 in which any two vertices at distance at most two have exactly µ common neighbors.

On antipodal distance-regular graphs with μ = 1

On antipodal distance-regular graphs with μ = 1

Classification of the family [formula omitted] of antipodal tight graphs

Let Γ be an antipodal distance-regular graph with diameter 4 and eigenvalues θ 0 > θ 1 > θ 2 > θ 3 > θ 4 . Then its Krein parameter q 11 4 vanishes precisely when Γ is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when Γ is locally strongly regular with nontrivial eigenvalues p : = θ 2 and − q : = θ 3 . When this is the case, the intersection parameters of Γ can be parameterized by p, q and the size of the antipodal classes r of Γ, hence we denote Γ by AT 4 ( p , q , r ) . Jurišić conjectured that the AT 4 ( p , q , r ) family is finite and that, aside from the Conway–Smith graph, the Soicher2 graph and the 3 . Fi 24 − graph, all graphs in this family have parameters belonging to one of the following four subfamilies: ( i ) q | p , r = q , ( ii ) q | p , r = 2 , ( iii ) p = q − 2 , r = q − 1 , ( iv ) p = q − 2 , r = 2 . In this paper we settle the first subfamily. Specifically, we show that for a graph AT 4 ( qs , q , q ) there are exactly five possibilities for the pair ( s , q ) , with an example for each: the Johnson graph J ( 8 , 4 ) for ( 1 , 2 ) , the halved 8-cube for ( 2 , 2 ) , the 3 . O 6 − ( 3 ) graph for ( 1 , 3 ) , the Meixner2 graph for ( 2 , 4 ) and the 3 . O 7 ( 3 ) graph for ( 3 , 3 ) . The fact that the μ-graphs of the graphs in this subfamily are completely multipartite is very crucial in this paper.

On graphs with complete multipartite [formula omitted]-graphs

Jurišić and Koolen proposed to study 1-homogeneous distance-regular graphs, whose μ -graphs (that is, the graphs induced on the common neighbours of two vertices at distance 2) are complete multipartite. Examples include the Johnson graph J (8, 4), the halved 8-cube, the known generalized quadrangle of order (4, 2), an antipodal distance-regular graph constructed by T. Meixner and the Patterson graph. We investigate a more general situation, namely, requiring the graphs to have complete multipartite μ -graphs, and that the intersection number α exists, which means that for a triple ( x , y , z ) of vertices in Γ , such that x and y are adjacent and z is at distance 2 from x and y , the number α ( x , y , z ) of common neighbours of x , y and z does not depend on the choice of a triple. The latter condition is satisfied by any 1-homogeneous graph. Let K t × n denote the complete multipartite graph with t parts, each of which consists of an n -coclique. We show that if Γ is a graph whose μ -graphs are all isomorphic to K t × n and whose intersection number α exists, then α = t , as conjectured by Jurišić and Koolen, provided α ≥ 2 . We also prove t ≤ 4 , and that equality holds only when Γ is the unique distance-regular graph 3 . O 7 ( 3 ) .

Lattices associated with distance-regular graphs

Let L be a finite set associated with cliques of a distance-regular graph of order ( s , t ) , with d -cliques of Johnson graphs and antipodal distance-regular graphs of diameter d , respectively. If we partially order L by the ordinary inclusion, three families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials.

Distance-regular graphs with complete multipartite μ-graphs and AT4 family

Let Γ be an antipodal distance-regular graph of diameter 4, with eigenvalues $$\theta_0>\theta_1>\theta_2>\theta_3>\theta_4$$ . Then its Krein parameter $$q_{11}^4$$ vanishes precisely when Γ is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when Γ is locally strongly regular with nontrivial eigenvalues $$p:=\theta_2$$ and $$-q:=\theta_3$$ . When this is the case, the intersection parameters of Γ can be parametrized by p, q and the size of the antipodal classes r of Γ. Let Γ be an antipodal tight graph of diameter 4, denoted by AT4 (p, q, r), and let the μ-graph be a graph that is induced by the common neighbours of two vertices at distance 2. Then we show that all the μ-graphs of Γ are complete multipartite if and only if Γ is AT4(sq,q,q) for some natural number s. As a consequence, we derive new existence conditions for graphs of the AT4 family whose μ-graphs are not complete multipartite. Another interesting application of our results is also that we were able to show that the μ-graphs of a distance-regular graph with the same intersection array as the Patterson graph are the complete bipartite graph K 4,4.

Siamese objects, and their relation to color graphs, association schemes and Steiner designs

Our motivation stems from the paper [9], in which the authors introduce a classof decompositions of complete graphs into certain strongly regular graphs whichshare a common spread. Analyzing their construction, we soon came to realize thatthese strongly regular graphs are GQ(s,t)-graphs, and thus, due to a famous resultof A.E. Brouwer, deletion of the spread leads in each case to an antipodal distanceregular graph which is an (s+1)-fold cover of the complete graph K

Relative difference sets fixed by inversion and Cayley graphs

Using graph theoretical technique, we present a construction of a ( 30 , 2 , 29 , 14 ) -relative difference set fixed by inversion in the smallest finite simple group—the alternating group A 5 . To our knowledge this is the first example known of relative difference sets in the finite simple groups with a non-trivial forbidden subgroup. A connection is then established between some relative difference sets fixed by inversion and certain antipodal distance-regular Cayley graphs. With the connection, several families of antipodal distance-regular Cayley graphs which are coverings of complete graphs are presented.