Distance-regular graphs with classical parameters that support a uniform structure: case q≤1q \le 1q≤1 (original) (raw)
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arXiv (Cornell University), 2023
Let Γ = (X, R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set R. Fix a vertex x ∈ X, and define R f = R \ {yz | ∂(x, y) = ∂(x, z)}, where ∂ denotes the path-length distance in Γ. Observe that the graph Γ f = (X, R f) is bipartite. We say that Γ supports a uniform structure with respect to x whenever Γ f has a uniform structure with respect to x in the sense of Miklavič and Terwilliger [7]. Assume that Γ is a distance-regular graph with classical parameters (D, q, α, β) and diameter D ≥ 4. Recall that q is an integer such that q ∈ {−1, 0}. The purpose of this paper is to study when Γ supports a uniform structure with respect to x. We studied the case q ≤ 1 in [3], and so in this paper we assume q ≥ 2. Let T = T (x) denote the Terwilliger algebra of Γ with respect to x. Under an additional assumption that every irreducible T-module with endpoint 1 is thin, we show that if Γ supports a uniform structure with respect to x, then either α = 0 or α = q, β = q 2 (q D − 1)/(q − 1), and D ≡ 0 (mod 6).
A characterization of bipartite distance-regular graphs
Linear Algebra and its Applications, 2014
It is well-known that the halved graphs of a bipartite distanceregular 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 distanceregular.
On the existence of certain distance-regular graphs
Journal of Combinatorial Theory, Series B, 1982
Distance-regular graphs of valency > 2, diameter m, and girth 2m with the additional property that any two points having maximal distance belong to a unique 2m circuit are investigated. It is shown that such graphs can exist only if m < 3; if m = 3 only a finite number of valencies prove to be feasible.
Algebraic characterizations of bipartite distance-regular graphs
2011
Bipartite graphs are combinatorial objects bearing some interesting symmetries. Thus, their spectra—eigenvalues of its adjacency matrix—are symmetric about zero, as the corresponding eigenvectors come into pairs. Moreover, vertices in the same (respectively, different) independent set are always at even (respectively, odd) distance. Both properties have well-known consequences in most properties and parameters of such graphs. Roughly speaking, we could say that the conditions for a given property to hold in a general graph can be somehow relaxed to guaranty the same property for a bipartite graph. In this paper we comment upon this phenomenon in the framework of distance-regular graphs for which several characterizations, both of combinatorial or algebraic nature, are known. Thus, the presented characterizations of bipartite distance-regular graphs involve such parameters as the numbers of walks between vertices (entries of the powers of the adjacency matrix A), the crossed local mu...
Pseudo 1-homogeneous distance-regular graphs
Journal of Algebraic Combinatorics, 2008
Let be a distance-regular graph of diameter d ≥ 2 and a 1 = 0. Let θ be a real number. A pseudo cosine sequence for θ is a sequence of real numbers σ 0 ,. .. , σ d such that σ 0 = 1 and c i σ i−1 + a i σ i + b i σ i+1 = θσ i for all i ∈ {0,. .. , d −1}. Furthermore, a pseudo primitive idempotent for θ is E θ = s d i=0 σ i A i , where s is any nonzero scalar. Letv be the characteristic vector of a vertex v ∈ V. For an edge xy of and the characteristic vector w of the set of common neighbours of x and y, we say that the edge xy is tight with respect to θ whenever θ = k and a nontrivial linear combination of vectors Ex, Eŷ and Ew is contained in Span{ẑ | z ∈ V , ∂(z, x) = d = ∂(z, y)}. When an edge of is tight with respect to two distinct real numbers, a parameterization with d + 1 parameters of the members of the intersection array of is given (using the pseudo cosines σ 1 ,. .. , σ d , and an auxiliary parameter ε). Let S be the set of all the vertices of that are not at distance d from both vertices x and y that are adjacent. The graph is pseudo 1-homogeneous with respect to xy whenever the distance partition of S corresponding to the distances from x and y is equitable in the subgraph induced on S. We show is pseudo 1-homogeneous with respect to the edge xy if and only if the edge xy is tight with respect to two distinct real numbers. Finally, let us fix a vertex x of. Then the graph is pseudo 1-homogeneous with respect to any edge xy, and the local graph of x is connected if and only if there is the above parameterization with d + 1 parameters σ 1 ,. .. , σ d , ε and the local graph of x is strongly regular with nontrivial eigenvalues a 1 σ/(1 + σ) and (σ 2 − 1)/(σ − σ 2).
The parameters of bipartite Q-polynomial distance-regular graphs
Journal of Algebraic Combinatorics, 2002
Let Γ denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ 0 , θ 1 , ..., θ D is a Q-polynomial ordering of the eigenvalues of Γ. This sequence is known to satisfy the recurrence θ i−1 −βθ i +θ i+1 = 0 (0 < i < D), for some real scalar β. Let q denote a complex scalar such that q + q −1 = β. Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.
On complementary distance pattern uniform graphs
International Journal of Applied Research, 2019
It was Koshy (2010) who introduced and investigated the concept Complementary Distance Pattern Uniform (CDPU) sets in a connected graph. In this paper, the researcher introduces one variety of Complementary Distance Pattern Uniform (CDPU) graphs which is α Complementary Distance Pattern Uniform (αcdpu) graphs. A couple of results are generated in this study. Some of which are the following: α(G + H) ≤ min {|V(G)|,|V(H)|} where G and H be connected graphs. Let G be a connected graph and H be a disconnected graph. Then α (G + H) ≤ |V(G)|. Let G be a self-centered graph and H be any graph. Then α (G • H) = |V(G)|. Let Knbe a complete graph of order n and H be any graph. Then α (Kn• H) = n. Let Cn be a cycle of order n and H be any graph. Then α (Cn•H)=n. Let G and H be graphs with isolated vertices u ϵ V(G) and v ϵ V(H). Then αu (G+H) = 2. Let K1,n, and Km,nbe a star of order n +1 and a complete bipartite graph of order n + m, respectively.
Distance–regular graphs having the M -property
Linear and Multilinear Algebra, 2012
We analyze when the Moore-Penrose inverse of the combinatorial Laplacian of a distanceregular graph is an M -matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore-Penrose inverse of the combinatorial Laplacian of a distance-regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance-regular graph has the M -property. We prove that only distance-regular graphs with diameter up to three can have the M -property and we give a characterization of the graphs that satisfy the M -property in terms of their intersection array. Moreover, we exhaustively analyze strongly regular graphs having the M -property and we give some families of distance regular graphs with diameter three that satisfy the M -property. Roughly speaking, we prove that all distance-regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance-regular graphs with diameter three, and no distance-regular graphs with diameter greater than three, have the M -property. In addition, we conjecture that no primitive distance-regular graph with diameter three has the M -property.
Distance-regular graphs with or at least half the valency
Journal of Combinatorial Theory, Series A, 2012
In this paper, we study the distance-regular graphs Γ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of Γ. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter three or a line graph.