Bipartite Distance-regular Graph Research Articles

Distance-regular Cayley graphs over dicyclic groups

The characterization of distance-regular Cayley graphs originates from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a partial classification of distance-regular Cayley graphs on dicyclic groups is obtained. More specifically, it is shown that every distance-regular Cayley graph on a dicyclic group is a complete graph, a complete multipartite graph, or a non-antipodal bipartite distance-regular graph with diameter 3 satisfying some additional conditions.

On Bipartite Distance-Regular Cayley Graphs with Small Diameter

We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group — the part of this bipartite Cayley graph which contains the identity element — and $\mathbb{Z}_{2}$. We apply this to the case of bipartite distance-regular Cayley graphs with diameter three, and consider cases where the sufficient conditions are not satisfied for some specific groups such as the dihedral group.We also extend a result by Miklavič and Potočnik that relates difference sets to bipartite distance-regular Cayley graphs with diameter three to the case of diameter four. This new case involves certain partial geometric difference sets and — in the antipodal case — relative difference sets.

A combinatorial basis for Terwilliger algebra modules of a bipartite distance-regular graph

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 for any integer i, let Γi(x) denote the set of vertices at distance i from x. Let V=ℂX denote the vector space over ℂ consisting of column vectors whose coordinates are indexed by X and whose entries are in ℂ, and for z∈X let ẑ denote the element of V with a 1 in the z coordinate and 0 in all other coordinates. Fix vertices x,u,v where u∈Γ2(x) and v∈Γ2(x)∩Γ2(u), and let T=T(x) denote the Terwilliger algebra with respect to x. Under certain additional combinatorial assumptions, we give a combinatorially-defined spanning set for a T-module of endpoint 2,and we give the action of the adjacency matrix on this spanning set. The vectors in our spanning set are defined as sums and differences of vectors ẑ, where the vertices z are chosen based on the their distances from x,u, and v.We use this T-module to construct combinatorially-defined bases for all isomorphism classes of irreducible T-modules of endpoint 2 for examples including the Doubled Odd graphs, the Double Hoffman–Singleton graph, Tutte’s 12-cage graph, and the Foster graph. We provide a list of several other graphs satisfying our conditions.

Extremal problems on distance spectra of graphs

Motivated by recent progresses on spectral extremal graph theory, in this paper, we aim to investigate the existence of cycles with given length in a graph in terms of its distance spectral radius. First of all, we show that if G is a connected bipartite graph with λ1(D(G))≤λ1(D(K1,n−1)), then G contains a C4 unless G≅K1,n−1. When n is sufficiently large with respect to k, as a corollary, we show that Sk(D(G))≥2n−2k if G is a C4-free bipartite graph. Besides, we prove that Sk(D(G))≥2n−2k if G is a bipartite distance regular graph. These two results partially solve a problem proposed by Lin (2019). Secondly, we give sufficient conditions for the existence of a Hamilton cycle or Hamilton path in a balanced bipartite graph in terms of the distance spectral radius.

On the Terwilliger algebra of certain family of bipartite distance-regular graphs with Δ_2 = 0

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 Ai (0 ≤ i ≤ D) denote the distance matrices of Γ. We abbreviate A := A1. 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. Fix x ∈ X and 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 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 study the structure of irreducible T-modules of endpoint 2. Let W denote an irreducible T-module with endpoint 2, and let v denote a nonzero vector in E2*W. We show that W = span({Ei*Ai − 2E2*v ∣ 2 ≤ i ≤ D} ∪ {Ei*Ai + 2E2*v ∣ 2 ≤ i ≤ D − 2}). It turns out that, except for a particular family of bipartite distance-regular graphs with D = 5, this result is already known in the literature. Assume now that Γ is a member of this particular family of graphs. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2 and it is not thin. We give a basis for this T-module.

Merging the first and third classes in bipartite distance-regular graphs

Merging the first and third classes in a connected graph is the operation of adding edges between all vertices at distance 3 in the original graph while keeping the original edges. We determine when merging the first and third classes in a bipartite distance-regular graph produces a distance-regular graph.

A spectral version of the Moore problem for bipartite regular graphs

Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta< 2\sqrt{k-1}$. Our bound gives the exact value of $b(k,\theta)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $\theta$, diameter $d$ and girth $g$ such that $g\geq 2d-2$. For certain values of $d$, there are infinitely many such graphs of various valencies $k$. However, for $d=11$ or $d\geq 15$, we prove that there are no bipartite distance-regular graphs with $g\geq 2d-2$.

On the metric dimension of incidence graphs

A resolving set for a graph Γ 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 dimensionμ(Γ) is the smallest size of a resolving set for Γ. We consider the metric dimension of two families of incidence graphs: incidence graphs of symmetric designs, and incidence graphs of symmetric transversal designs (i.e. symmetric nets). These graphs are the bipartite distance-regular graphs of diameter 3, and the bipartite, antipodal distance-regular graphs of diameter 4, respectively. In each case, we use the probabilistic method in the manner used by Babai to obtain bounds on the metric dimension of strongly regular graphs, and are able to show that μ(Γ)=O(nlogn) (where n is the number of vertices).

On Bipartite $Q$-Polynomial Distance-Regular Graphs with Diameter 9, 10, or 11

Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D$. In [Caughman (2004)], Caughman showed that if $D \ge 12$, then $\Gamma$ is $Q$-polynomial if and only if one of the following (i)-(iv) holds: (i) $\Gamma$ is the ordinary $2D$-cycle, (ii) $\Gamma$ is the Hamming cube $H(D,2)$, (iii) $\Gamma$ is the antipodal quotient of the Hamming cube $H(2D,2)$, (iv) the intersection numbers of $\Gamma$ satisfy $c_i = (q^i - 1)/(q-1)$, $b_i = (q^D-q^i)/(q-1)$ $(0 \le i \le D)$, where $q$ is an integer at least $2$. In this paper we show that the above result is true also for bipartite distance-regular graphs with $D \in \{9,10,11\}$.

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.

An A-invariant subspace for bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.

Distance-regular graphs with small number of distinct distance eigenvalues

In this paper we characterize distance-regular graphs with diameter three having exactly three distinct distance eigenvalues, and also bipartite distance-regular graphs with diameter four having three distinct distance eigenvalues. We derive some properties and give particular examples of such graphs. We also present an infinite family of bipartite semiregular graphs with diameter four having exactly four distinct distance eigenvalues. With these results, we address some problems posed in Atik and Panigrahi (2015) [3].

On bipartite distance-regular graphs with exactly one non-thin [formula omitted]-module with endpoint two

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.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. An irreducible T-module W is said to be thin whenever dim(Ei∗W)≤1 for 0≤i≤D. By the endpoint of an irreducible T-module W we mean min{i|Ei∗W≠0}.Fix x∈X and assume that Γ has, up to isomorphism, exactly one irreducible T-module W with endpoint 2, and that W is non-thin with dim(E2∗W)=1, dim(ED−1∗W)≤1 and dim(Ei∗W)≤2 for 3≤i≤D. We prove that for 2≤i≤D, there exist complex scalars αi,βi such that |Γi−1(x)∩Γi−1(y)∩Γ1(z)|=αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)| for all y∈Γ2(x) and z∈Γi(x)∩Γi(y). Furthermore, we prove Δ2=0 and either D=5 or c2∈{1,2}. We show there exist integers 3≤f≤ℓ≤D−2 such that dim(Ei∗W)=2 if and only if f≤i≤ℓ.

On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint two

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 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. An irreducible T-module W is said to be thin whenever dim Ei⁎W≤1 for 0≤i≤D. By the endpoint of W we mean min{i|Ei⁎W≠0}. For 0≤i≤D, let Γi(z) denote the set of vertices in X that are distance i from vertex z. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. In this paper we prove the following are equivalent: (i) Δ2>0 and for 2≤i≤D−2 there exist complex scalars αi,βi with the following property: for all x,y,z∈X such that ∂(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)|; (ii) For all x∈X there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra T(x) with endpoint two, and these modules are thin.

On the Terwilliger algebra of bipartite distance-regular graphs with [formula omitted] and [formula omitted

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. It is known 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}.We find the structure of irreducible T-modules of endpoint 2 for graphs Γ which have 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)|, in case when Δ2=0 and c2=2. The case when Δ2=0 and c2=1 is already studied by MacLean et al. [15].We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2 and it is not thin. We give a basis for this T-module, and we give the action of A on this basis.

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.

The local eigenvalues of a bipartite distance-regular graph

We consider a bipartite distance-regular graph Γ with vertex set X, diameter D≥4, and valency k≥3. For 0≤i≤D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. We assume there exist scalars r,s,t∈R, not all zero, such that r|Γ1(x)∩Γ1(y)∩Γ2(z)|+s|Γ2(x)∩Γ2(y)∩Γ1(z)|+t=0 for all x,y,z∈X with path-length distances ∂(x,y)=2,∂(x,z)=3,∂(y,z)=3. Fix x∈X, and let Γ22 denote the graph with vertex set X̃={y∈X∣∂(x,y)=2} and edge set R̃={yz∣y,z∈X̃,∂(y,z)=2}. We show that the adjacency matrix of the local graph Γ22 has at most four distinct eigenvalues. We are motivated by the fact that our assumption above holds if Γ is Q-polynomial.

Bipartite distance-regular graphs: The [formula omitted]-polynomial property and pseudo primitive idempotents

Let Γ denote a bipartite distance-regular graph with diameter at least 4 and valency at least 3. Fix a vertex of Γ and let T denote the corresponding Terwilliger algebra. Suppose that Γ is Q-polynomial and there are two non-isomorphic irreducible T-modules with endpoint 2. We show that, unless the intersection numbers of Γ fit one exceptional case (which is not known to correspond to an actual graph), the entry-wise product of pseudo primitive idempotents associated with these modules is a linear combination of two pseudo primitive idempotents. This result relates to a conjecture of MacLean and Terwilliger.

A characterization of bipartite Leonard pairs using the notion of a tail

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A⁎:V→V that satisfy (i) and (ii) below.(i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A⁎ is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A⁎ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Very roughly speaking, a Leonard pair is a linear algebraic abstraction of a Q-polynomial distance-regular graph. There is a well-known class of distance-regular graphs said to be bipartite and there is a related notion of a bipartite Leonard pair. Recently, M.S. Lang introduced the notion of a tail for bipartite distance-regular graphs and there is an abstract version of this tail notion. Lang characterized the bipartite Q-polynomial distance-regular graphs using tails. In this paper, we obtain a similar characterization of the bipartite Leonard pairs using tails. Whereas Lang's arguments relied on the combinatorics of a distance-regular graph, our results are purely algebraic in nature.

A characterization of bipartite distance-regular graphs

It is well-known that the halved graphs of a bipartite distance-regular graph are distance-regular. Examples are given to show that the converse does not hold. Thus, a natural question is to find out when the converse is true. In this paper we give a quasi-spectral characterization of a connected bipartite weighted 2-punctually distance-regular graph whose halved graphs are distance-regular. In the case the spectral diameter is even we show that the graph characterized above is distance-regular.