Distance-regular Graphs Research Articles

Distance-regular Cayley graphs with least eigenvalue -2

We classify the distance-regular Cayley graphs with least eigenvalue -2 and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.

Distance mean-regular graphs

We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let $$\Gamma $$Γ be a graph with vertex set V, diameter D, adjacency matrix $$\varvec{A}$$A, and adjacency algebra $$\mathcal{A}$$A. Then, $$\Gamma $$Γ is distance mean-regular when, for a given $$u\in V$$uźV, the averages of the intersection numbers $$p_{ij}^h(u,v)=|\Gamma _i(u)\cap \Gamma _j(v)|$$pijh(u,v)=|Γi(u)źΓj(v)| (number of vertices at distance i from u and distance j from v) computed over all vertices v at a given distance $$h\in \{0,1,\ldots ,D\}$$hź{0,1,ź,D} from u, do not depend on u. In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of $$\Gamma $$Γ and, hence, they generate a subalgebra of $$\mathcal{A}$$A. Some other algebras associated to distance mean-regular graphs are also characterized.

Distance-Regular Graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.

On the distance spectra of graphs

The distance matrix of a graph G is the matrix containing the pairwise distances between vertices. The distance eigenvalues of G are the eigenvalues of its distance matrix and they form the distance spectrum of G. We determine the distance spectra of double odd graphs and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.

Arithmetic completely regular codes

In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.

The Nonexistence of a Distance-Regular Graph with Intersection Array {22,16,5;1,2,20}

We prove that a distance-regular graph with intersection array {22,16,5;1,2,20} does not exist. To prove this, we assume that such a graph exists and derive some combinatorial properties of its local graph. Then we construct a partial linear space from the local graph to display the contradiction.

On the Terwilliger algebra of bipartite distance-regular graphs with Δ2 = 0 and c2 = 1

Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and let A denote the adjacency matrix of Γ. For x∈X and for 0≤i≤D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. We first show that Δ2=0 implies that D≤5 or c2∈{1,2}.For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A,E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D, Ei⁎ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W we mean min{i|Ei⁎W≠0}.In this paper we assume Γ has the property that for 2≤i≤D−1, there exist complex scalars αi, βi such that for all x,y,z∈X with ∂(x,y)=2, ∂(x,z)=i, ∂(y,z)=i, we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|. We additionally assume that Δ2=0 with c2=1.Under the above assumptions we study the algebra T. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2. We give an orthogonal basis for this T-module, and we give the action of A on this basis.

On automorphisms of distance-regular graphs with intersection arrays {2r + 1, 2r − 2, 1; 1, 2, 2r + 1}

On automorphisms of distance-regular graphs with intersection arrays {2r + 1, 2r − 2, 1; 1, 2, 2r + 1}

On Automorphisms of Distance-Regular Graph with Intersection Array $$\{18,15,9;\,1,1,10\}$$ { 18 , 15 , 9 ; 1 , 1 , 10 }

Recently, Makhnev and Nirova found intersection arrays of distance-regular graphs with \(\lambda =2\) and at most 4096 vertices. In the case of primitive graphs of diameter 3 with \(\mu = 1\) there corresponding arrays are \(\{18,15,9;1,1,10\}\), \(\{33,30,8;1,1,30\}\) or \(\{39,36,4;1,1,36\}\). In this work, possible orders and subgraphs of fixed points of the hypothetical distance-regular graph with intersection array \(\{18,15,9;1,1,10\}\) are studied. In particular, graph with intersection array \(\{18,15,9;1,1,10\}\) is not vertex symmetric.

The spectral excess theorem for distance-regular graphs having distance-d 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 $d$ 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 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.

To the theory of [formula omitted]-ary Steiner and other-type trades

We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner T(k−1,k,v) bitrades, extended 1-perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-regular with certain parameters. As an application of the results, we find the minimum cardinality of q-ary Steiner Tq(k−1,k,v) bitrades and show a connection of minimum such bitrades with dual polar subgraphs of the Grassmann graph Jq(v,k).

Quotient-polynomial graphs

As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex u induces the same regular partition around u, where all vertices of each cell are equidistant from u. Some properties and characterizations of such graphs are studied. For instance, all quotient-polynomial graphs are walk-regular and distance-polynomial. Also, we show that every quotient-polynomial graph generates a (symmetric) association scheme.

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.

A collection of results concerning electric resistance and simple random walk on distance-regular graphs

In this paper, we prove a number of related results on distance-regular graphs concerning electric resistance and simple random walk. We begin by proving several results on electric resistance; in particular we prove a sharp constant bounding the ratio of electrical resistances between any two pairs of points and give a counterexample to a conjecture made in a previous paper regarding the growth of resistances with respect to distance. We then show how a number of strong bounds on moments of hitting times, cover times, and related quantities for simple random walk may be deduced from the bound on resistance.

The spectral excess theorem for distance-regular graphs having distance-d graph with fewer distinct eigenvalues

Let Γ be a distance-regular graph with diameter d and Kneser graph K=Γd, the distance-d graph of Γ. We say that Γ is partially antipodal when K has fewer distinct eigenvalues than Γ. 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 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 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.

The distance spectrum of corona and cluster of two graphs

Let G be a connected graph with a distance matrix D. The D-eigenvalues {μ1,μ2,…,…,μp} of G are the eigenvalues of D and form the distance spectrum or D-spectrum of G. Given two graphs G with vertex set {v1,v2,……,vp} and H, the corona G∘H is defined as the graph obtained by taking p copies of H and for each i, joining the ith vertex of G to all the vertices in the ith copy of H. Let H be a rooted graph rooted at u. Then the cluster G{H} is defined as the graph obtained by taking p copies of H and for each i, joining the ith vertex of G to the root in the ith copy of H. In this paper we describe the distance spectrum of G∘H, for a connected distance regular graph G and any r-regular graph H in terms of the distance spectrum of G and adjacency spectrum of H. We also describe the distance spectrum of G{Kn}, where G is a connected distance regular graph.

On Middle Cube Graphs

We study a family of graphs related to the $n$-cube. The middle cube graph of parameter k is the subgraph of $Q_{2k-1}$ induced by the set of vertices whose binary representation has either $k-1$ or $k$ number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors).

Some Results on the Eigenvalues of Distance-Regular Graphs

In 1986, Terwilliger showed that there is a strong relation between the eigenvalues of a distance-regular graph and the eigenvalues of a local graph. In particular, he showed that the eigenvalues of a local graph are bounded in terms of the eigenvalues of a distance-regular graph, and he also showed that if an eigenvalue $$\theta $$? of the distance-regular graph has multiplicity m less than its valency k, then $$-1- \frac{b_1}{\theta +1}$$-1-b1?+1 is an eigenvalue for any local graph with multiplicity at least $$k-m$$k-m. In this paper, we are going to generalize the results of Terwilliger to a broader class of subgraphs. Instead of local graphs, we consider induced subgraphs of distance-regular graphs (and more generally t-walk-regular graphs with t a positive integer) which satisfies certain regularity conditions. Then we apply the results to obtain bounds on eigenvalues of t-walk-regular graphs if the girth equals 3, 4, 5, 6 or 8. In particular, we will show that the second largest eigenvalue of a distance-regular graph with girth 6 and valency k is at most $$k-1$$k-1, and we will show that the only such graphs having $$k-1$$k-1 as its second largest eigenvalue are the doubled Odd graphs.

Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3

The study of graphs with the property specified in the title has earlier been reduced to the case of neighborhoods with parameters (35, 18, 9, 9), (36, 21, 12, 12), (40, 27, 18,18), (50, 28, 15, 16), (56, 45, 36, 36), and (64, 27, 10, 12). It is proved that a completely regular graph in which the neighborhoods of vertices are strongly regular graphs with the corresponding parameters is either a strongly regular graph with parameters (76, 35, 18, 14) or (64, 35, 18, 20) or a Taylor graph with intersection array {35, 16, 1; 1, 16, 35} (this is the main theorem). As a consequence, a classification of distance regular graphs in which the neighborhoods of vertices are strongly regular graphs with a non principal eigenvalue r, 2 < r ≤ 3, is obtained.

Leonard triples, the Racah algebra, and some distance-regular graphs of Racah type

By a Leonard triple, we mean a triple of diagonalizable operators on a finite-dimensional vector space such that for each operator, there is an ordering of an eigenbasis for the selected operator with respect to which the other two operators are irreducible tridiagonal.Let C denote the field of complex numbers and let D denote an integer at least 3. Let 12H″(2D+1,2) denote the halved graph of the (2D+1)-cube with respect to the original P-polynomial structure R0,R1,…,RD and another Q-polynomial structure E0,E2,E4,…,E3,E1 in terms of the original ones. Let 12H¯(4D,2) denote the folded halved graph of the 4D-cube and let 12H¯(4D+2,2) denote the folded halved graph of the (4D+2)-cube. Note that they are all distance-regular graphs of Racah type.In this paper we consider the relations between the above three graphs and the Leonard triples or the Racah algebra over C. Our results are described as follows.1. Fix a vertex of 12H″(2D+1,2) and let T1 denote the corresponding Terwilliger algebra with respect to this vertex. We first construct three elements U1, U1⁎ and U1ε of T1. Then we show that the triple U1,U1⁎,U1ε acts on each irreducible T1-module as a Leonard triple. Moreover, let K1 be a Racah algebra with its generators and real parameters satisfying certain conditions. We display a C-algebra homomorphism from K1 to T1.2. Fix a vertex of 12H¯(4D,2) and let T2 denote the Terwilliger algebra of 12H¯(4D,2) with respect to this vertex. We construct three elements U2, U2⁎, U2ε of T2 and show that the triple U2,U2⁎,U2ε not only acts on each irreducible T2-module as a Leonard triple but also satisfies some very appealing equations. Moreover, let W denote an irreducible T2-module with type ψ and let Kψ be a Racah algebra with respect to ψ. Then there exists a Kψ-module structure on W.3. Fix a vertex of 12H¯(4D+2,2) and let T3 denote the Terwilliger algebra of 12H¯(4D+2,2) with respect to this vertex. We construct three elements U3,U3⁎,U3ε of T3 and show that the triple U3,U3⁎,U3ε not only acts on each irreducible T3-module as a Leonard triple but also satisfies some very appealing equations. Moreover, let W denote an irreducible T3-module with auxiliary parameter e and let Ke be a Racah algebra with respect to e. Then there exists a Ke-module structure on W.