Distance-regular Graphs Research Articles

Non-Binary Diameter Perfect Constant-Weight Codes

Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weight anticodes. Each one of these families of diameter perfect codes raises some different questions. We consider some of these questions and leave lot of ground for further research. Finally, as a consequence, some t-intersecting families related to the well-known Erdos-Ko-Rado theorem, are constructed.

Self-orthogonal codes from equitable partitions of association schemes

We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method, we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained codes are optimal. Further, we introduce a notion of self-orthogonal subspace codes. We show that under some conditions equitable partitions of association schemes yield such self-orthogonal subspace codes and we give some examples from distance-regular graphs.

Distance-regular graphs admitting a perfect 1-code

In this paper, we study the problem that which of distance-regular graphs admit a perfect 1-code. Among other results, we characterize distance-regular line graphs which admit a perfect 1-code. Moreover, we characterize all known distance-regular graphs with small valency at most 4, the distance-regular graphs with known putative intersection arrays for valency 5, and all distance-regular graphs with girth 3 and valency 6 or 7 which admit a perfect 1-code.

Study of some properties of complement of open subset inclusion graph of a topological space

In the recent paper, authors introduced a graph topological structure, called as open subset inclusion graph 𝚥(𝜏) of a topological space (𝑋, 𝜏) on a finite set 𝑋 and discussed some important properties of this graph. In this paper, we discuss some properties of the graph 𝚥(𝜏) 𝑐 . It is shown that, if 𝜏 is a discrete topology defined on nonempty set 𝑋 with |𝑋| ≤ 3, then the graph 𝚥(𝜏) 𝑐 is bipartite, and if |𝑋| = 2, then the graph 𝚥(𝜏) 𝑐 is regular & complete bipartite. Moreover, if 𝜏 is a discrete topology defined on nonempty set 𝑋 with |𝑋| = 2 or |𝑋| = 3 then it is shown that the graph 𝚥(𝜏) 𝑐 is Hamiltonian, vertex-transitive, edge-transitive and has a perfect matching. We also provide exact value of the independence number, vertex connectivity and edge connectivity of the graph 𝚥(𝜏) 𝑐 of a discrete topology defined on nonempty set𝑋 with |𝑋| = 2 or |𝑋| = 3. Main finding of this work is that, if (𝑋, 𝜏) is a discrete topological space with |𝑋| = 2 or |𝑋| = 3 then it is shown that 𝚥(𝜏) 𝑐 is distance-transitive graph and distance regular graph.

Некоторые графы Шилла с b = 5 не существуют

A Shilla graph is a distance-regular graph of diameter 3 that has a second eigenvalue equal to a = a3 . Koolen and Park found admissible arrays of intersections of the Shill graphs with b = 3 (there were 12 of them). Belousov I.N. found feasible intersection arrays of the Shilla graphs with b = 4 (there were 50 of them) and b = 5 (there were 82 of them). It is proved in the paper that distance-regular Schill graphs with b = 5 and intersection arrays {305,248,62;1,2,244}, {315,256,64;1,2,252}, {345,280,64;1,4,276},{615,496,124;1,4,492}, {815,656,164;1,2,652}, {855,688,172;1,4,684}, {855,688,170;1,5,684}, {910,732,180;1,10, 728}, {1000,804,201;1,3,800}, {1045,840,210;1,6,836}, {1055,848,212;1,4,844}, {1080,868,215;1,5,864}, {1155,928,232;1,2,924}, {1185,952,245;1,5,948}, {1235,992,248;1,8,988},{1535,1232,308;1,8,1228}, {1560,1252,310;1,10,1248}, {1615,1296,324;1,12,1292}, {1665,1336,334;1,2,1332} do not exist.

SHILLA GRAPHS WITH \(b=5\) AND \(b=6\)

A \(Q\)-polynomial Shilla graph with \(b = 5\) has intersection arrays \(\{105t,4(21t+1),16(t+1); 1,4 (t+1),84t\}\), \(t\in\{3,4,19\}\). The paper proves that distance-regular graphs with these intersection arrays do not exist. Moreover, feasible intersection arrays of \(Q\)-polynomial Shilla graphs with \(b = 6\) are found.

Thin Distance-Regular Graphs with Classical Parameters $(D,q,q, \frac{q^{t}-1}{q-1}-1)$ with $t> D$ are the Grassmann Graphs

In the survey paper by Van Dam, Koolen and Tanaka (2016), they asked to classify the thin $Q$-polynomial distance-regular graphs. In this paper, we show that a thin distance-regular graph with the same intersection numbers as a Grassmann graph $J_q(n, D)~ (n \geqslant 2D)$ is the Grassmann graph if $D$ is large enough.

The 2-partially distance-regular graphs such that their second largest local eigenvalues are at most one

In this paper, we will classify the 2-partially distance-regular graphs whose each local graph has second largest eigenvalue at most one. As an application, we classify the 2-walk-regular graphs with an eigenvalue −b12−1. These results generalize the results of Koolen and Yu for distance-regular graphs.

On the distance spectrum of minimal cages and associated distance biregular graphs

A (k,g)-cage is a k-regular simple graph of girth g with minimum possible number of vertices. In this paper, (k,g)-cages which are Moore graphs are referred as minimal (k,g)-cages. A simple connected graph is called distance regular (DR) if all its vertices have the same intersection array. A bipartite graph is called distance biregular (DBR) if all the vertices of the same partite set admit the same intersection array. It is known that minimal (k,g)-cages are DR graphs and their subdivisions are DBR graphs. In this paper, for minimal (k,g)-cages we give a formula for distance spectral radius in terms of k and g, and also determine polynomials of degree ⌊g2⌋, which is the diameter of the graph. This polynomial gives all distance eigenvalues when the variable is substituted by adjacency eigenvalues. We show that a minimal (k,g)-cage of diameter d has d+1 distinct distance eigenvalues, and this partially answers a problem posed in [1]. We prove that every DBR graph is a 2-partitioned transmission regular graph and then give a formula for its distance spectral radius. By this formula we obtain the distance spectral radius of subdivision of minimal (k,g)-cages. Finally we determine the full distance spectrum of subdivision of some minimal (k,g)-cages.

Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters

We determine the possible scaling limits in the quantum central limit theorem with respect to the Gibbs state, for a growing distance-regular graph that has so-called classical parameters with base unequal to one. We also describe explicitly the corresponding weak limits of the normalized spectral distribution of the adjacency matrix. We demonstrate our results with the known infinite families of distance-regular graphs having classical parameters and with unbounded diameter.

Distance-regular Terwilliger graphs with intersection arrays {50,42,1;1,2,50} and {50,42,9;1,2,42} do not exist

Distance-regular Terwilliger graphs with intersection arrays {50,42,1;1,2,50} and {50,42,9;1,2,42} do not exist

On nonexistence of distance regular graphs with the intersection array ${53,40,28,16;1,4,10,28}$

On nonexistence of distance regular graphs with the intersection array ${53,40,28,16;1,4,10,28}$

On the Nonexistence of Distance-Regular Graphs with Intersection Array $$ \{ 53,40,28,16; 1,4,10,28\}$$

On the Nonexistence of Distance-Regular Graphs with Intersection Array $$ \{ 53,40,28,16; 1,4,10,28\}$$

A characterization of Johnson and Hamming graphs and proof of Babai's conjecture

One of the central results in the representation theory of distance-regular graphs classifies distance-regular graphs with μ≥2 and second largest eigenvalue θ1=b1−1. In this paper we give a classification under the (weaker) approximate eigenvalue constraint θ1≥(1−ε)b1 for the class of geometric distance-regular graphs. As an application, we confirm Babai's conjecture on the minimal degree of the automorphism group of distance-regular graphs of bounded diameter. This conjecture asserts that if X is a primitive distance-regular graph with n vertices, and X is not a Hamming graph or a Johnson graph, then the automorphism group Aut(X) has minimal degree ≥cn for some constant c>0. It follows that Aut(X) satisfies strong structural constraints.

Inverse Problems in the Theory of Distance-Regular Graphs: Dual 2-Designs

Inverse Problems in the Theory of Distance-Regular Graphs: Dual 2-Designs

Totally bipartite tridiagonal pairs

There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or totally bipartite (TB). Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey--Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey--Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; and (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair.

Norton Algebras of the Hamming Graphs via Linear Characters


 
 
 The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.
 
 

On a metric property of perfect colorings

Given a perfect coloring of a graph, we prove that the $L_1$ distance between two rows of the adjacency matrix of the graph is not less than the $L_1$ distance between the corresponding rows of the parameter matrix of the coloring. With the help of an algebraic approach, we deduce corollaries of this result for perfect $2$-colorings, perfect colorings in distance-$l$ graphs and in distance-regular graphs. We also provide examples when the obtained property reject several putative parameter matrices of perfect colorings in infinite graphs.

Designs in Finite Metric Spaces: A Probabilistic Approach

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in Q-polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel function in approximation theory is given. As an application we derive limit laws on the weight distributions of binary orthogonal arrays of strength going to infinity. An analogous result for combinatorial designs of strength going to infinity is given.

Edge-disjoint spanning trees and forests of graphs

The spanning tree packing number of a graph G is the maximum number of edge-disjoint spanning trees in G, and the arboricity of G is the minimum number of edge-disjoint forests needed to partition the edge set of G. In this paper, we give bounds on the spanning tree packing number and the arboricity of graphs in terms of effective resistances. As applications, we show that equiarboreal graphs are uniformly dense, and determine the maximum number of edge-disjoint spanning c-forests of equiarboreal graphs, including edge-transitive graphs, 1-walk-regular graphs, distance-regular graphs and so on. The arboricity of a regular graph can be derived from our effective resistance bounds directly.