Antipodal Distance-regular Graphs Of Diameter Research Articles

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

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

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.

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.

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.

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.

AT4 family and 2-homogeneous graphs

Let Γ denote an antipodal distance-regular graph of diameter four, with eigenvalues k= θ 0> θ 1>⋯> θ 4 and antipodal class size r. Then its Krein parameters satisfy q 11 2 q 12 3 q 13 4 q 22 2 q 22 4 q 23 3 q 24 4 q 33 4>0, q 12 2=q 12 4=q 14 4=q 22 3=q 23 4=q 34 4=0 and q 11 1,q 11 3,q 13 3,q 33 3∈(r−2) R +. It remains to consider only two more Krein bounds, namely q 11 4⩾0 and q 44 4⩾0. Jurišić and Koolen showed that vanishing of the Krein parameter q 11 4 of Γ implies that Γ is 1-homogeneous in the sense of Nomura, so it is also locally strongly regular. We study vanishing of the Krein parameter q 44 4 of Γ. In this case a well-known result of Cameron et al. implies that Γ is locally strongly regular. We gather some evidence that vanishing of the Krein parameter q 44 4 implies Γ is either triangle-free (in which case it is 1-homogeneous) or the Krein parameter q 11 4 vanishes as well. Then we prove that the vanishing of both Krein parameters q 11 4 and q 44 4 of Γ implies that every second subconstituent graph is again an antipodal distance-regular graph of diameter four. Finally, if Γ is also a double-cover, i.e., r=2, i.e., Q-polynomial, then it is 2-homogeneous in the sense of Nomura.

Krein parameters and antipodal tight graphs with diameter 3 and 4

We determine which Krein parameters of nonbipartite antipodal distance-regular graphs of diameter 3 and 4 can vanish, and give combinatorial interpretations of their vanishing. We also study tight distance-regular graphs of diameter 3 and 4. In the case of diameter 3, tight graphs are precisely the Taylor graphs. In the case of antipodal distance-regular graphs of diameter 4, tight graphs are precisely the graphs for which the Krein parameter q 11 4 vanishes.

Homotopy in Q-polynomial distance-regular graphs

Let Γ denote a Q-polynomial distance-regular graph with diameter d⩾3. We show that if the valency is at least three, then the intersection number p 12 3 is at least two; consequently the girth is at most six. We then consider a condition on the dual eigenvalues of Γ that must hold if Γ is the quotient of an antipodal distance-regular graph of diameter D⩾7; we call Γ a pseudoquotient whenever this condition holds. For our main result, we show that if Γ is not a pseudoquotient, then any cycle in Γ can be ‘decomposed’ into cycles of length at most six. We present this result using homotopy.

Antipodal distance-regular graphs of diameter four and five

An antipodal distance-regular graph of diameter four or five is a covering graph of a connected strongly regular graph. We give existence conditions for these graphs and show for some types of strongly regular graphs that no nontrivial covers exist. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 69–77, 1999

The Spectra of Complementary Subgraphs in a Strongly Regular Graph

As an application of Jacobi's identity on complementary minors, one can exhibit a simple explicit relation between the characteristic polynomials of any pair of complementary induced subgraphs in a strongly regular graph. In particular, there is an explicit relation between the spectra of the first and second subconstituents with respect to any vertex. Several consequences will be presented; for example, there follow rather strong parametric restrictions on an SRG that has some second subconstituent isomorphic to an antipodal distance-regular graph of diameter three. The latter class of graphs also admits a palatable complementary-minors formula; this will be stated, together with an application.

Second neighbourhoods of strongly regular graphs

Gardiner, A.D., C.D. Godsil, A.D. Hensel and G.F. Royle, Second neighbourhoods of strongly regular graphs, Discrete Mathematics 103 (1992) 161–170. Some antipodal distance-regular graphs of diameter three arise as the graph induced by the vertices at distance two from a given vertex in a strongly regular graph. We show that if every vertex in a strongly regular graph G has this property, then G is the noncollinearity graph of a special type of semipartial geometry. As these semipartial geometries have all been classified, we obtain a list of the antipodal distance-regular graphs of diameter three that can arise in this way.

Antipodal graphs of diameter three

Distance-regular graphs of diameter three are of three (almost distinct) kinds: primitive, bipartite, and antipodal. An antipodal graph of diameter three is just an r-fold covering of a complete graph K k+1 for some r⩽ k. Its intersection array and spectrum are determined by the parameters r, k together with the number c of 2-arcs joining any two vertices at distance two. Most such graphs have girth three. In this note we consider antipodal distance-regular graphs of diameter three and girth ⩾ 4. If r=2, then the only graphs are “ K k+1, k+1 minus a 1-factor.” We therefore assume r⩾3. The graphs with c=1 necessarily have r= k and were classified in lsqb3rsqb. We prove the inequality r−2>c 1 2 (Theorem 2), list the feasible parameter sets when c=2 or 3 (Corollary 1), and conclude that every 3-fold or 4-fold antipodal covering of a complete graph has girth three (Corollary 2).