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 A₅. 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.     

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.     

The distance-regular graphs such that all of its second largest local eigenvalues are at most one

In this paper, we classify distance regular graphs such that all of its second largest local eigenvalues are at most one. Also we discuss the consequences for the smallest eigenvalue of a distance-regular graph. These extend a result by the first author, who classified the distance-regular graphs with smallest eigenvalue .     

Arc-transitive distance-regular covers of cliques with <Emphasis Type="Italic">λ</Emphasis> = 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.     

Distance-regular graphs of large diameter that are completely regular clique graphs

A connected graph is said to be a completely regular clique graph with parameters (s, c), \(s, c \in {\mathbb {N}}\), if there is a collection \(\mathcal {C}\) of completely regular cliques of size \(s+1\) such that every edge is contained in exactly c members of \(\mathcal {C}\). It is known that many families of distance-regular graphs are completely regular clique graphs. In this paper, we determine completely regular clique graph structures, i.e., the choices of \(\mathcal {C}\), of all known families of distance-regular graphs with unbounded diameter. In particular, we show that all distance-regular graphs in this category are completely regular clique graphs except the Doob graphs, the twisted Grassmann graphs and the Hermitean forms graphs. We also determine parameters (s, c); however, in a few cases we determine only s and give a bound on the value c. Our result is a generalization of a series of works by J. Hemmeter and others who determined distance-regular graphs in this category that are bipartite halves of bipartite distance-regular graphs.     

Nonexistence of some Antipodal Distance-regular Graphs of Diameter Four

We find an inequality involving the eigenvalues of a regular graph; equality holds if and only if the graph is strongly regular. We apply this inequality to the first subconstituents of a distance-regular graph and obtain a simple proof of the fundamental bound for distance-regular graphs, discovered by Juri i , Koolen and Terwilliger. Using this we show that for distance-regular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular. From these results we prove the nonexistence of distance-regular graphs associated to 20 feasible intersection arrays from the book Distance-Regular Graphs by Brouwer, Cohen and Neumaier .     

On Subgraphs in Distance-Regular Graphs

Terwilliger [15] has given the diameter bound d (s – 1)(k – 1) + 1 for distance-regular graphs with girth 2s and valency k. We show that the only distance-regular graphs with even girth which reach this bound are the hypercubes and the doubled Odd graphs. Also we improve this bound for bipartite distance-regular graphs. Weichsel [17] conjectures that the only distance-regular subgraphs of a hypercube are the even polygons, the hypercubes and the doubled Odd graphs and proves this in the case of girth 4. We show that the only distance-regular subgraphs of a hypercube with girth 6 are the doubled Odd graphs. If the girth is equal to 8, then its valency is at most 12.     

The spectral excess theorem for distance-regular graphs having distance-<Emphasis Type="Italic">d</Emphasis> 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 \(d+1\) 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 more 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.     

On distance-regular graphs with fixed valency,II

Let be a distance-regular graph of valencyk 3 and diameterd. Suppose the intersection array hast columns different from ^t (1, 0,k – 1). Then it is shown thatd is bounded from above by a certain functionf(k, t) depending only onk andt. As an application, this theorem eliminates certain cubic distance-regular graphs to complete the classification of such graphs by Biggs et al.Supported in part by NSF grant MCS-8301826 and by the British SERC grant.     

Integral Cayley graphs over dihedral groups

In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group \(D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle \). Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over \(D_n\) in terms of the Boolean algebra of \(\langle a\rangle \), from which we find infinite classes of integral Cayley graphs over \(D_n\). In particular, we completely determine all integral Cayley graphs over the dihedral group \(D_p\) for a prime p.     

Normal cirulant graphs with noncyclic regular subroups

We prove that any circulant graph of order n with connection set S such that n and the order of ?(S), the subgroup of ? that fixes S set‐wise, are relatively prime, is also a Cayley graph on some noncyclic group, and shows that the converse does not hold in general. In the special case of normal circulants whose order is not divisible by 4, we classify all such graphs that are also Cayley graphs of a noncyclic group, and show that the noncyclic group must be metacyclic, generated by two cyclic groups whose orders are relatively prime. We construct an infinite family of normal circulants whose order is divisible by 4 that are also normal Cayley graphs on dihedral and noncyclic abelian groups. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On constructing normal and non-normal Cayley graphs

Bamberg and Giudici (2011) showed that the point graphs of certain generalised quadrangles of order $(q?1,q+1)$, where $q=p^k$ is a prime power with $p\ge 5$, are both normal and non-normal Cayley graphs for two isomorphic groups. We call these graphs BG-graphs. In this paper, we show that the Cayley graphs obtained from a finite number of BG-graphs by Cartesian product, direct product, and strong product also possess the property of being normal and non-normal Cayley graphs for two isomorphic groups.     

The non-orientable genus of some metacyclic groups

We describe non-orientable, octagonal embeddings for certain 4-valent, bipartite Cayley graphs of finite metacyclic groups, and give a class of examples for which this embedding realizes the non-orientable genus of the group. This yields a construction of Cayley graphs for which is arbitrarily large, where and are the orientable genus and the non-orientable genus of the Cayley graph.Work supported in part by the Research Council of Slovenia, Yugoslavia and NSF Contract DMS-8717441.Supported by NSF Contract DMS-8601760.     

Generalized Cayley graphs of semigroups I

We introduce the concept of generalized Cayley graphs of semigroups and discuss their fundamental properties, and then study a special case, the universal Cayley graphs of semigroups so that some general results are given and the universal Cayley graph of a -partial order of complete graphs with loops is described.     

The Cayley graphs of ℤ<Superscript>d</Superscript> and the limits of vertex-primitive graphs of <Emphasis Type="Italic">HA</Emphasis>-type

We study the limits of the finite graphs that admit some vertex-primitive group of automorphisms with a regular abelian normal subgroup. It was shown in [1] that these limits are Cayley graphs of the groups ?^d. In this article we prove that for each d > 1 the set of Cayley graphs of ?^d presenting the limits of finite graphs with vertex-primitive and edge-transitive groups of automorphisms is countable (in fact, we explicitly give countable subsets of these limit graphs). In addition, for d < 4 we list all Cayley graphs of ?^d that are limits of minimal vertex-primitive graphs. The proofs rely on a connection of the automorphism groups of Cayley graphs of ?^d with crystallographic groups.     

Antipodal covers of distance-transitive generalized polygons

In an earlier paper, the first two authors found all distance-regular antipodal covers of all known primitive distance-transitive graphs of diameter at least 3 with one possible exception. That remaining case is resolved here with the proof that a primitive and distance-transitive collinearity graph of a finite generalized 2d-gon with \(d\ge 3\) has no distance-regular antipodal cover of diameter 2d.     

Vertex reconstruction in Cayley graphs

In this survey paper we review recent results on the vertex reconstruction problem (which is not related to Ulam’s problem) in Cayley graphs. The problem of reconstructing an arbitrary vertex x from its r-neighbors, that are, vertices at distance at most r from x, consists of finding the minimum restrictions on the number of r-neighbors when such a reconstruction is possible. We discuss general results for simple, regular and Cayley graphs. To solve this problem for given Cayley graphs, it is essential to investigate their structural and combinatorial properties. We present such properties for Cayley graphs on the symmetric group and the hyperoctahedral group (the group of signed permutations) and overview the main results for them. The choice of generating sets for these graphs is motivated by applications in coding theory, computer science, molecular biology and physics.