Families Of Distance-regular Graphs Research Articles

The least Euclidean distortion constant of a distance-regular graph

In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin’s conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph of diameter d , the most contracted pairs of vertices are those at distance d . In this paper, we confirm Vallentin’s conjecture for several families of distance-regular graphs. We also provide counterexamples to this conjecture, where the largest contraction occurs between pairs of vertices at distance d − 1 . We suggest three alternative conjectures and prove them for various families of distance-regular graphs and for distance-regular graphs of diameter 3.

Of shadows and gaps in spatial search

Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty \etal ({\em Physical Review A}, {\bf 102}:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula.

Thin trees in some families of distance-regular graphs

ABSTRACT For some families of distance-regular graphs such as strongly regular graphs, Johnson graphs, cocktail-party graphs, and Hamming graphs, we give a polynomial-time construction of spanning trees of maximum degree that are thin trees under some mild assumptions on the parameters of the graph.

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.

Asymptotic Delsarte cliques in distance-regular graphs

We give a new bound on the parameter $$\lambda $$ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph G, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014. arXiv:1409.3041 ). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai et al. 2013). The proof is based on a clique geometry found by Metsch (Des Codes Cryptogr 1(2):99–116, 1991) under certain constraints on the parameters. We also give a simplified proof of the following asymptotic consequence of Metsch’s result: If $$k\mu = o(\lambda ^2)$$ , then each edge of G belongs to a unique maximal clique of size asymptotically equal to $$\lambda $$ , and all other cliques have size $$o(\lambda )$$ . Here k denotes the degree and $$\mu $$ the number of common neighbors of a pair of vertices at distance 2. We point out that Metsch’s cliques are “asymptotically Delsarte” when $$k\mu = o(\lambda ^2)$$ , so families of distance-regular graphs with parameters satisfying $$k\mu = o(\lambda ^2)$$ are “asymptotically Delsarte-geometric.”

Resolving sets for four families of distance-regular graphs

Abstract Recently upper bounds on the metric dimension of Grassmann graphs, bilinear forms graphs, doubled Grassmann graphs and twisted Grassmann graphs have been obtained by Bailey, Meagher, Feng and the present authors. In this paper, we continue this research and construct resolving sets with smaller sizes for these graphs. As a result, we improve the known upper bounds on the metric dimension for the latter three graphs.

New families of completely regular codes and their corresponding distance regular coset graphs

In this paper three new infinite families of linear binary completely regular codes are constructed. They have covering radius ? = 3 and 4, and are halves of binary Hamming and binary extended Hamming codes of length n = 2 m ?1 and 2 m , where m is even. There are also shown some combinatorial (binomial) identities which are new, to our knowledge.These completely regular codes induce, in the usual way, i.e., as coset graphs, three infinite families of distance-regular graphs of diameter three and four. This description of such graphs is new.

Two distance-regular graphs

We construct two families of distance-regular graphs, namely the subgraph of the dual polar graph of type B_3(q) induced on the vertices far from a fixed point, and the subgraph of the dual polar graph of type D_4(q) induced on the vertices far from a fixed edge. The latter is the extended bipartite double of the former.

A new family of distance-regular graphs with unbounded diameter

We construct distance-regular graphs with the same – classical – parameters as the Grassmann graphs on the e-dimensional subspaces of a (2e+1)-dimensional space over an arbitrary finite field. This provides the first known family of non-vertex-transitive distance-regular graphs with unbounded diameter.

A New Family of Distance-regular Graphs with Unbounded Diameter

We construct distance-regular graphs with the same-classical-parameters as the Grassman graphs on the e-dimensional subspaces of a (2e+1)-dimensional space over an arbitrary finite field. This provides the first known family of non-transitive distance-regular graphs with unbounded diameter.

Cycle-balance conditions for distance-regular graphs

In a distance-regular graph, the partition with respect to distance from a vertex supports a unique eigenvector for each eigenvalue. There may be non-singleton vertex sets whose corresponding distance partition also supports eigenvectors. We consider the members of three families of distance regular graphs, the Johnson Graphs, Hamming Graphs and Complete Multipartite graphs. For each we determine all such sets which support an eigenvector for the next to largest eigenvalue. These sets exhibit the underlying geometric structure of the graph.

A Family of Antipodal Distance-Regular Graphs Related to the Classical Preparata Codes

A new family of distance-regular graphs is constructed. They are antipodal 22t−1-fold covers of the complete graph on 22t vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs. There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matricess and their underlying 3-class association scheme is formally dual to the scheme of a system of linked symmetric designs obtained from Kerdock sets of skew matrices in characteristic two.

The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes

Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals (1974), and Delsarte-Goethals (1975). It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z/sub 4/, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z/sub 4/ domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z/sub 4/-and the Nordstrom-Robinson code is self-dual-which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z/sub 4/-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z/sub 4/, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the I(Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z/sub 4/, but extended Hamming codes of length n/spl ges/32 and the Golay code are not. Using Z/sub 4/-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new family of distance-regular graphs and the {0, 1, 2}-cliques in dual polar graphs

Let G be a dual polar graph of type B d ( q) or C d ( q). Define a graph Gˆ on the same vertex set by having vertices x and y adjacent in Gˆ if they are of distance 1 or 2 in G. The graph Gˆ was recently shown to be distance-regular [10, 12]. In fact, Gˆ is the halved graph of a bipartite distance-regular graph. When G is of type C d ( q) and q is odd, this bipartite graph is new.