Distance-regular Graph Of Diameter Research Articles

On a class of coedge regular graphs

We study graphs in which for every edge and all -subgraphs are 2-cocliques. We give a description of connected edge-regular graphs for . In particular, the following examples confirm that the inequality is a sharp bound for strong regularity: the -gon, the icosahedron graph, the graph in and the distance-regular graph of diameter 4 with intersection massive , which is an antipodal 3-covering of the strongly regular graph with parameters .

Classification of three-class association schemes using backtracking with dynamic variable ordering

We describe several techniques for the exhaustive computer generation of non-isomorphic association schemes with a given set of intersection numbers using a backtracking algorithm with forward checking and dynamic variable ordering. We have applied these techniques to the classification of certain open parameter sets for three-class association schemes listed by Van Dam in (Three-class association schemes, J. Algebraic Combin. 10 (1999) 69–107) for which we present several new results. Among these are some new (imprimitive) distance regular graphs of diameter 3.

The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph

Let ? be a distance-regular graph of diameter D. Let X denote the vertex set of ? and let Y be a nonempty subset of X. We define an algebra ? = ?(Y). This algebra is finite dimensional and semisimple. If Y consists of a single vertex then ? is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible ?-modules. We define endpoints and thin condition on irreducible ?-modules as a generalization of the case when Y consists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector of Y, it is thin if and only if Y is a completely regular code of ?. By considering a suitable subset Y, every irreducible ?(x)-module of endpoint i can be regarded as an irreducible ?(Y)-module of endpoint 0.

The geometric girth of a distance-regular graph having certain thin irreducible modules for the Terwilliger algebra

Let Γ be a distance-regular graph of diameter D . Suppose that Γ does not have any induced subgraph isomorphic to K 2 , 1 , 1 . In this case, the length of a shortest reduced circuit in Γ is called the geometric girth gg ( Γ ) of Γ . Except for ordinary polygons, all known examples have a property that gg ( Γ ) ≤ 12 in general, and gg ( Γ ) ≤ 8 if a 1 ≠ 0 . Is there an absolute constant bound on the geometric girth of a distance-regular graph with valency at least three? This is one of the main problems in the field of distance-regular graphs. P. Terwilliger defined an algebra T = T ( x ) with respect to a base vertex x , which is called a subconstituent algebra or a Terwilliger algebra. The investigation of irreducible T -modules and their thin property proved to be a very important tool to study structures of distance-regular graphs. B. Collins proved that if every irreducible T -module is thin then gg ( Γ ) is at most 8, and if gg ( Γ ) = 8 , then a 1 = 0 and Γ is a generalized octagon. In this paper, we prove the same result under an assumption that every irreducible T -module of endpoint at most 3 is thin.

Applications of the retracing method for distance-regular graphs

In the previous paper (J. Combin. Theory Ser. B 79 (2000) 211) we introduced the retracing method for distance-regular graphs and gave some applications. In this paper, we give other applications of this method. In particular, we prove the following result: Theorem. Let Γ be a distance-regular graph of diameter d with r=|{ i∣( c i , a i , b i )=( c 1, a 1, b 1)}|≥2 and c r+1 ≥2. Let m, s and t be positive integers with s≤ m, m+ t≤ d and ( s, t)≠(1,1). Suppose b m− s+1 =⋯= b m =1+ b m+1 , c m+1 =⋯= c m+ t =1+ c m and a m− s+2 =⋯= a m+ t−1 =0. Then the following hold. (1) If b m+1 ≥2, then t≤ r−2[ s/3]. (2) If c m ≥2, then s≤ r−2[ t/3].

A distance-regular graph with bipartite geodetically closed subgraphs

Let Γ be a distance-regular graph of diameter d, and t be an integer with 2≤ t≤ d−1 such that a t−1 =0. For any pair ( u, v) of vertices, let Π( u, v) be the subgraph induced by the vertices lying on shortest paths between u and v. We prove that if Π( u, v) is a bipartite geodetically closed subgraph for some pair ( u, v) of vertices at distance t, then Π( x, y) is a bipartite geodetically closed subgraph for any pair ( x, y) of vertices at distance less than or equal to t. In particular, Π( x, y) is either a path, an ordinary polygon, a hyper cube or a projective incidence graph.

On a Distance-Regular Graph of Even Height with k e = k f

Let Γ be a distance-regular graph of diameter d. The height of Γ is defined by h = max{j∣p d d,j ≠ 0}. Let e, f be positive integers such that e < f and e + f≤d, and let d = 2e + s for some positive integer s. We show that if k e = k f , h≤ 2s and the height h is even, then Γ is an antipodal 2-cover.

On Strongly Regular Graphs withμ = 1

We consider strongly regular graphs in which each non-adjacent pair of vertices has exactly one common neighbour. These graphs give rise to partial linear spaces (one of which is a partial quadrangle) and a distance-regular graph of diameter three. The lower bound for the valency of the graph in terms of the number of common neighbours of two adjacent vertices is derived.

A Distance-Regular Graph with Strongly Closed Subgraphs

Let Γ be a distance-regular graph of diameter d, valency k and r := maxi | (c i,b i) = (c 1,b 1). Let q be an integer with r + 1 ≤ q ≤ d − 1.In this paper we prove the following results: Theorem 1 Suppose for any pair of vertices at distance q there exists a strongly closed subgraph of diameter q containing them. Then for any integer i with 1 ≤ i ≤ q and for any pair of vertices at distance i there exists a strongly closed subgraph of diameter i containing them. Theorem 2 If r ≥ 2, then c 2r+3 ≠ 1.As a corollary of Theorem 2 we have d ≤ k 2(r + 1) if r ≥ 2.

Distance-regular Graphs of the Height h

The height of a distance-regular graph of the diameter d is defined by h=max{j|pdd,j≠0}. We show that if Γ is a distance-regular graph of diameter d, height h>1 and every pdd,h-graph is a clique, then h∈{d−1,d}.

Spectral Characterization of Some Generalized Odd Graphs

Suppose G is a connected, k-regular graph such that SpecOGUaSpecOGU where G is a distance-regular graph of diameter d with parameters a1a a2aa adˇ1a 0 and ad > 0; i.e., a generalized odd graph, we show that G must be distance-regular with the same intersection array as that of G in terms of the notion of Homan Polynomials. Fur- thermore, G is isomorphic to G if G is one of the odd polygon C2da1, the Odd graph Oda1, the folded O2da 1U-cube, the coset graph of binary Golay code Oda 3U, the Homan- Singleton graphOda 2U, the Gewirtz graphOda 2U, the Higman-Sims graphOda 2U ,o r the second subconstituent of the Higman-Sims graphOda 2U.

Some Formulas for Spin Models on Distance-Regular Graphs

A spin model is a square matrix W satisfying certain conditions which ensure that it yields an invariant of knots and links via a statistical mechanical construction of V. F. R. Jones. Recently F. Jaeger gave a topological construction for each spin model W of an association scheme which contains W in its Bose–Mesner algebra. Shortly thereafter, K. Nomura gave a simple algebraic construction of such a Bose–Mesner algebra N ( W ). In this paper we study the case W ∈ A ⊆ N ( W ), where A is the Bose–Mesner algebra of a distance-regular graph. We show the following results. Let Γ =( X , R ) be a distance-regular graph of diameter d >1 such that the Bose–Mesner algebra A of Γ satisfies W ∈ A ⊆ N ( W ) for some spin model W on X . Write W =∑ d i =0 t i A i , where A i denotes the i th adjacency matrix. Set x i = t −1 i −1 t i and p = x −1 1 x 2 . Then x i = p i −1 x 1 holds for all i . Moreover, the eigenvalues and the intersection numbers of Γ are rational functions of x 1 and p .

Antipodal distance-regular graphs of diameter four and five

An antipodal distance-regular graph of diameter four or five is a covering graph of a connected strongly regular graph. We give existence conditions for these graphs and show for some types of strongly regular graphs that no nontrivial covers exist. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 69–77, 1999

The Spectra of Complementary Subgraphs in a Strongly Regular Graph

As an application of Jacobi's identity on complementary minors, one can exhibit a simple explicit relation between the characteristic polynomials of any pair of complementary induced subgraphs in a strongly regular graph. In particular, there is an explicit relation between the spectra of the first and second subconstituents with respect to any vertex. Several consequences will be presented; for example, there follow rather strong parametric restrictions on an SRG that has some second subconstituent isomorphic to an antipodal distance-regular graph of diameter three. The latter class of graphs also admits a palatable complementary-minors formula; this will be stated, together with an application.

Crooked Functions, Bent Functions, and Distance Regular Graphs

Let $V$ and $W$ be $n$-dimensional vector spaces over $GF(2)$. A mapping $Q:V\rightarrow W$ is called crooked if it satisfies the following three properties: $Q(0)=0$; $Q(x)+Q(y)+Q(z)+Q(x+y+z)\neq 0$ for any three distinct $x,y,z$; $Q(x)+Q(y)+Q(z)+Q(x+a)+Q(y+a)+Q(z+a)\neq 0$ if $a\neq 0$ ($x,y,z$ arbitrary). We show that every crooked function gives rise to a distance regular graph of diameter 3 having $\lambda=0$ and $\mu=2$ which is a cover of the complete graph. Our approach is a generalization of a recent construction found by de Caen, Mathon, and Moorhouse. We study graph-theoretical properties of the resulting graphs, including their automorphisms. Also we demonstrate a connection between crooked functions and bent functions.

Distance-Regular Graphs with c i = b {d-i} and Antipodal Double Covers

Let G be a distance-regular graph of diameter d and valency k > 2. Suppose there exists an integers with d ≤ 2s such that c_i e b_{d-i} for all 1 ≤ i ≤ s. Then G is an antipodal double cover.

Distance-regular graphs, MH-colourings and MLD-colourings

It is shown that the only bipartite distance regular graphs of diameter 3 which are MLD-colourable are the incidence graphs of complementary Hadamard designs.

Distance-regular Subgraphs in a Distance-regular Graph, III

Let Γ be a distance-regular graph with l(1, a 1, b 1)=1 and c s + 1=1 for some positive integer s.We show the existence of a certain distance-regular graph of diameter s,containing given two vertices at distance s,as a subgraph in Γ.

Twice Q-Polynomial Distance-Regular Graphs

Let Γ be a distance-regular graph of diameter at least three. Suppose Γ is Q-polynomial with respect to distinct eigenvalues θ and ψ of Γ . We calculate the kite number f 2 j ( x , y , z ) from the intersection numbers of Γ and the dual eigenvalues to θ and ψ . This implies that the kite numbers f ij ( x , y , z ) are independent of the vertices x , y , z . We use this result to show that the vertex neighborhood graphs of Γ are strongly regular and we calculate the parameters of these graphs from the kite numbers f 2 j and the intersection numbers of Γ .

Distance-Regular Graphs withbt=1 and Antipodal Double-Covers

Let Λ be a distance-regular graph of diameter d and valency k >2. If b t =1 and 2 t ⩽ d , then Λ is an antipodal double-cover. Consequently, if f >2 is the multiplicity of an eigenvalue of the adjacency matrix of Λ and if Λ is not an antipodal double-cover then d ⩽2 f −3. This result is an improvement of Godsil's bound.