Distance-regular Graphs Research Articles

On the $Q$-Polynomial Property of the Full Bipartite Graph of a Hamming Graph

The $Q$-polynomial property is an algebraic property of distance-regular graphs, that was introduced by Delsarte in his study of coding theory. Many distance-regular graphs admit the $Q$-polynomial property. Only recently the $Q$-polynomial property has been generalized to graphs that are not necessarily distance-regular. In [J. Combin. Theory Series. A, 205:105872, 2024] it was shown that the graphs arising from the Hasse diagrams of the so-called attenuated space posets are $Q$-polynomial. These posets could be viewed as $q$-analogs of the Hamming posets, which were not studied in that paper. The main goal of this article is to fill this gap by showing that the graphs arising from the Hasse diagrams of the Hamming posets are $Q$-polynomial.

Dual Bipartite $Q$-Polynomial Distance-Regular Graphs and Dual Uniform Structures

Let $\Gamma$ denote a dual bipartite $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D \geq 3$. Fix $x \in X$, and let $L^*$ and $R^*$ denote the corresponding dual lowering and dual raising matrix, respectively. We show that a certain linear dependency among $R^* L^{* 2}, L^* R^* L^*, L^{* 2} R^*, L^*$ holds, and determine whether this linear dependency endow $\Gamma$ with a dual uniform or dual strongly uniform structure. Precisely, except for two special cases a dual uniform structure is always attained, and except for four special cases a dual strongly uniform structure is always attained.

M-Distance-regular graphs and their relation to multivariate P-polynomial association schemes

An association scheme is P-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate P-polynomial association schemes of type (α,β) were introduced by Bernard et al., and multivariate P-polynomial association schemes were later defined by Bannai et al. In this paper, the notion of m-distance-regular graph is defined and shown to give a graph interpretation of the multivariate P-polynomial association schemes. Various examples are provided. Refined structures and additional constraints for multivariate P-polynomial association schemes and m-distance-regular graphs are also considered. In particular, bivariate P-polynomial schemes of type (α,β) are discussed, and their connection to 2-distance-regular graphs is established.

Bivariate Q-polynomial structures for the nonbinary Johnson scheme and the association scheme obtained from attenuated spaces

The study of P-polynomial association schemes (distance-regular graphs) and Q-polynomial association schemes, and in particular P- and Q-polynomial association schemes, has been a central theme not only in the theory of association schemes but also in the whole study of algebraic combinatorics in general. Leonard's theorem (1982) says that the spherical functions (or the character tables) of P- and Q-polynomial association schemes are described by Askey-Wilson orthogonal polynomials or their relatives. These polynomials are one-variable orthogonal polynomials. It seems that the new attempt to define and study higher rank P- and Q-polynomial association schemes had been hoped for, but had gotten only limited success. The first very successful attempt was initiated recently by Bernard-Crampé-d'Andecy-Vinet-Zaimi, and then followed by Bannai-Kurihara-Zhao-Zhu. The general theory and some explicit examples of families of higher rank (multivariate) P- and/or Q-polynomial association schemes have been obtained there. The main purpose of the present paper is to prove that some important families of association schemes are shown to be bivariate Q-polynomial. Namely, we show that all the nonbinary Johnson association schemes and all the association schemes obtained from attenuated spaces are bivariate Q-polynomial. It should be noted that the parameter restrictions needed in the previous papers are completely lifted in this paper. Our proofs are done by explicitly calculating the Krein parameters of these association schemes. At the end, we mention some speculations and indications of what we can expect in the future study.

On the (Non-)Existence of Tight Distance-Regular Graphs: a Local Approach

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 3.$ Jurišić and Vidali conjectured that if $\Gamma$ is tight with classical parameters $(D,b,\alpha,\beta)$, $b\geq 2$, then $\Gamma$ is not locally the block graph of an orthogonal array nor the block graph of a Steiner system. In the present paper, we prove this conjecture and, furthermore, extend it from the following aspect. Assume that for every triple of vertices $x, y, z$ of $\Gamma$, where $x$ and $y$ are adjacent, and $z$ is at distance $2$ from both $x$ and $y$, the number of common neighbors of $x$, $y$, $z$ is constant. We then show that if $\Gamma$ is locally the block graph of an orthogonal array (resp.~a Steiner system) with smallest eigenvalue $-m$, $m\geq 3$, then the intersection number $c_2$ is not equal to $m^2$ (resp. $m(m+1)$). Using this result, we prove that if a tight distance-regular graph $\Gamma$ is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of $\Gamma$ is bounded by a function in the parameter $b=b_1/(1+\theta_1)$, where $b_1$ is the intersection number of $\Gamma$ and $\theta_1$ is the second largest eigenvalue of $\Gamma$.

Distance-regular graphs with exactly one positive q-distance eigenvalue

In this paper, we study the q-distance matrix for a distance-regular graph and show that the q-distance matrix of a distance-regular graph with classical parameters (D,q,α,β) has exactly three distinct eigenvalues, of which one is zero. Moreover, we study distance-regular graphs whose q-distance matrix has exactly one positive eigenvalue.

Distance-regular graphs with a few q-distance eigenvalues

In this paper we study when the q-distance matrix of a distance-regular graph has few distinct eigenvalues. We mainly concentrate on diameter 3.

On the spectra and spectral radii of token graphs

Let G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document}, has as vertices the nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${n\\atopwithdelims ()k}$$\\end{document}k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(K_n)$$\\end{document} is the Johnson graph J(n, k), which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} in terms of the spectrum of G. For instance, when G is walk-regular, an exact value for the spectral radius ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} (or maximum eigenvalue) of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} is obtained. When G is distance-regular, other eigenvalues of its 2-token graph are derived using the theory of equitable partitions. A generalization of Aldous’ spectral gap conjecture (which is now a theorem) is proposed.

Об автоморфизмах графов с массивами пересечений {44,40,12;1,5,33} и {48,35,9;1,7,40}

Distance-regular graph Γ of diameter 3 with strongly regular graphs Γ2 and Γ3 has intersection array {r(c2+1)+a3, r c2, a3 + 1; 1, c2, r(c2 + 1)} (M.S. Nirova). For distance-regular graph Γ of diameter 3 and degree 44 there are exactly 7 feasible intersection arrays. For each of them graph Γ3 is strongly regular. For intersection array {44, 30, 5; 1, 3, 40} we have a3 = 4, c2 = 3, r = 10, Γ2 has parameters (540,440,358,360) and Γ3 has parameters (540,55,10,5). Graph does not exist (Koolen-Park). For intersection array {44, 35, 3; 1, 5, 42} graph Γ3 has parameters (375,22,5,1) and does not exist (its neighbourhood of vertex is the union of isolated 6-cliques). In this paper it is found futomorphisms of graphs with intersection arrays {44,40,12; 1,5,33 and 48,35,9; 1,7,40}.

The k-adjacency operators and adjacency Jacobi matrix on distance-regular graphs

The k-adjacency operators and adjacency Jacobi matrix on distance-regular graphs

Further study of distance-regular graphs with classical parameters with b < −1

Let Γ be a distance-regular graph of diameter d with classical parameters (d,b,α,β) with b<−1. Van Dam, Koolen and Tanaka [Distance-regular graphs, Electron. J. Combin. (2016) DS22] surveyed the classification of Γ. From this survey, the following four cases have not been studied: (1) d≥3, c2=a1=1; (2) d≥3, c2=1, a2=a1>1; (3) d=3, c2=1, a2>a1>1; (4) d=3, c2>1, a1≥1. Here ai, bi, ci (0≤i≤d) are the intersection numbers of Γ. In this paper, we study the above four cases. Our main results are as follows. For the case (1), Γ is the triality graph ▪; for the case (2), Γ is the collinearity graph of a generalized hexagon of order (a1+1,(a1+1)3). In particular, if a1+1 is a prime power, then the classical parameters of Γ are realized by the triality graphs ▪; for the case (3), Γ does not exist; for the case (4), precisely one of the following (i)–(iv) holds: (i) Γ is the dual polar graph ▪, where a1+1 is a prime power; (ii) Γ is the extended ternary Golay code graph; (iii) Γ is the large Witt graph M24; (iv) Γ is the collinearity graph of a maximal regular near hexagon with classical parameters (d,b,α,β)=(3,−a1−1,−c2+a1a1,(a1+1)(c2+a1+1)) with a1≥2, 2≤c2≤(a1)2+a1+1. In particular, if 2≤a1≤107−1, then c2=2 and a1∈{3,4,7,10,17,22,31,52,157}.

Distance-Regular Graphs with Classical Parameters that Support a Uniform Structure: Case $$q \le 1$$

Distance-Regular Graphs with Classical Parameters that Support a Uniform Structure: Case $$q \le 1$$

Spectra of quasi-strongly regular graphs

A quasi-strongly regular graph G of grade 2 with parameters (n,k,a;c1,c2) is a k-regular graph on n vertices such that every pair of adjacent vertices have a common neighbors, every pair of distinct nonadjacent vertices have c1 or c2 common neighbors, and for each ci(i=1,2), there exists a pair of non-adjacent vertices sharing ci common neighbors. The children G1 and G2 of the graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in G1 or G2 if and only if they share c1 or c2 neighbors, respectively. The graph G is a quasi-strongly regular graph of type A or B if it is of grade 2 and has a child that is a connected strongly regular graph or a disjoint union of cliques of order m(m>1), respectively. In this paper, we investigate the spectral properties of the graph G if it is a quasi-strongly regular graph of type A or B. Several examples of distance-regular graphs which are quasi-strongly regular graphs of type A are given. Moreover, we give several constructions of quasi-strongly regular graphs and calculate their spectrum.

On distance-regular graphs Γ of diameter 3 for which Γ 3 is a triangle-free graph

Abstract There exist well-known distance-regular graphs Γ of diameter 3 for which Γ 3 is a triangle-free graph. An example is given by the Johnson graph J (8, 3) with the intersection array {15, 8, 3;1, 4, 9}. The paper is concerned with the problem of the existence of distance-regular graphs Γ with the intersection arrays {78, 50, 9;1, 15, 60} and {174, 110, 18;1, 30, 132} for which Γ 3 is a triangle-free graph.

2-Reconstructibility of Strongly Regular Graphs and 2-Partially Distance-Regular Graphs

2-Reconstructibility of Strongly Regular Graphs and 2-Partially Distance-Regular Graphs

A Q-Polynomial Structure Associated with the Projective Geometry $$L_N(q)$$

There is a type of distance-regular graph, said to be Q-polynomial. In this paper, we investigate a generalized Q-polynomial property involving a graph that is not necessarily distance-regular. We give a detailed description of an example associated with the projective geometry $$L_N(q)$$ .

Combinatorial necessary conditions for regular graphs to induce periodic quantum walks

We derive combinatorial necessary conditions for discrete-time quantum walks defined by regular mixed graphs to be periodic. One useful necessary condition is that if a k-regular mixed graph with n vertices is periodic, then 2n/k must be an integer. As an application of this work, we determine periodicity of mixed complete graphs and mixed graphs with a prime number of vertices. Furthermore, we study periodicity of mixed strongly regular graphs and several classes of mixed distance-regular graphs, and extend existing results to mixed graphs.

A New Feasibility Condition for the AT4 Family

Let $\Gamma$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $\theta_0&gt;\theta_1&gt;\theta_2&gt;\theta_3&gt;\theta_4$. 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 $r=(p+q^3)(p+q)^{-1}$.

Correction to: Thin Q-Polynomial Distance-Regular Graphs Have Bounded $$c_{2}$$

Correction to: Thin Q-Polynomial Distance-Regular Graphs Have Bounded $$c_{2}$$

Tridiagonal pairs, alternating elements, and distance-regular graphs

The positive part Uq+ of Uq(slˆ2) has a presentation with two generators W0, W1 and two relations called the q-Serre relations. The algebra Uq+ contains some elements, said to be alternating. There are four kinds of alternating elements, denoted {W−k}k∈N, {Wk+1}k∈N, {Gk+1}k∈N, {G˜k+1}k∈N. The alternating elements of each kind mutually commute. A tridiagonal pair is an ordered pair of diagonalizable linear maps A,A⁎ on a nonzero, finite-dimensional vector space V, that each act in a (block) tridiagonal fashion on the eigenspaces of the other one. Let A, A⁎ denote a tridiagonal pair on V. Associated with this pair are six well-known direct sum decompositions of V; these are the eigenspace decompositions of A and A⁎, along with four decompositions of V that are often called split. In our main results, we assume that A, A⁎ has q-Serre type. Under this assumption A, A⁎ satisfy the q-Serre relations, and V becomes an irreducible Uq+-module on which W0=A and W1=A⁎. We describe how the alternating elements of Uq+ act on the above six decompositions of V. We show that for each decomposition, every alternating element acts in either a (block) diagonal, (block) upper bidiagonal, (block) lower bidiagonal, or (block) tridiagonal fashion. We investigate two special cases in detail. In the first case the eigenspaces of A and A⁎ all have dimension one. In the second case A and A⁎ are obtained by adjusting the adjacency matrix and a dual adjacency matrix of a distance-regular graph that has classical parameters and is formally self-dual.