117,948 research outputs found
Improving diameter bounds for distance-regular graphs
No full textIn this paper we study the sequence (c(i))(0 = 2j and cj > 1 then c(2j-1) > c(j) bolds. Using this we give improvements on diameter bounds by A. Hiraki, J.H. Koolen [An improvement of the Ivanov bound, Ann. Comb. 2 (2) (1998) 131-135], and L. Pyber [A bound for the diameter of distance-regular graphs, Combinatorica 19 (4) (1999) 549-553], respectively, by applying this inequality. (c) 2004 Elsevier Ltd. All rights reserved.X116sciescopu
Two theorems concerning the Bannai-Ito conjecture
No full textIn 1984 Bannai and Ito conjectured that there are finitely many distance-regular graphs with fixed valencies greater than two. In a series of papers, they showed that this is the case for valency 3 and 4, and also for the class of bipartite distance-regular graphs. To prove their result, they used a theorem concerning the intersection array of a triangle-free distance-regular graph, a theorem that was subsequently generalized in 1994 by Suzuki to distance-regular graphs whose intersection numbers satisfy a certain simple condition. More recently, Koolen and Moulton derived a more general version of Banni and Ito's theorem which they used to show that the Banai-Ito conjecture holds for valencies 5, 6 and 7, and which they subsequently extended to triangle-free distance-regular graphs in order to show that the Bannai-Ito conjecture holds for such graphs with valencies 8, 9 and 10. In this paper, we extend the theorems of Bannai and Ito, and Koolen and Moulton to arbitrary distance-regular graphs. (c) 2006 Elsevier Ltd. All rights reserved.X115sciescopu
A bound for the number of columns l(c,a,b) in the intersection array of a distance-regular graph
No full textIn this paper we give a bound for the number l(c,a,b) of columns (c,a,b)T in the intersection array of a distance-regular graph. We also show that this bound is intimately related to the Bannai–Ito conjecture
The vertex-connectivity of a distance-regular graph
No full textThe vertex-connectivity of a distance-regular graph equals its valency. (C) 2008 Dr Andries E. Brouwer. Published by Elsevier Ltd. All rights reserved.X1118sciescopu
Koolen de Vries Sundrome in the first adulthood patient of Southern India Ancestry
No full textKoolen-de Vries syndrome (KdVS, MIM#610443) is a rare malformation condition mainly characterized by cognitive impairment in association with craniofacial and visceral anomalies. The core phenotype is caused by mutations in the chromatin remodeler KANSL1 (MSL1V1, KIAA1267, KAT8 Regulatory NSL Complex Subunit 1, MIM#612452), which maps to 17q21.31 critical genomic region (Koolen et al., Nature Genetics 2012;44:639-641). Considering its molecular basis, KdVS is included in the group of Developmental Disorders of Chromatin Remodeling (DDCRs), also termed chromatinopathies. We describe the first KdVS patient of Southern India ethnicity, harboring the typical de novo 17q21.31 microdeletion, including KANSL1. Observed facial features and congenital anomalies are in line with the already reported KdVS phenotype, suggesting that phenotypic features are consistent across different ethnicities
Delsarte clique graphs
No full textIn this paper, we consider the class of Delsarte clique graphs, i.e. the class of distance-regular graphs with the property that each edge lies in a constant number of Delsarte cliques. There are many examples of Delsarte clique graphs such as the Hamming graphs, the Johnson graphs and the Grassmann graphs. Our main result is that, under mild conditions, for given s >= 2 there are finitely many Delsarte clique graphs which contain Delsarte cliques with size s + I. Further we classify the Delsarte clique graphs with small s. (c) 2005 Elsevier Ltd. All rights reserved.X1111sciescopu
The spectra of the local graphs of the twisted Grassmann graphs
No full textIn this paper we will determine the spectra of the local graphs of the twisted Grassmann graphs and look at its consequences for the associated Terwilliger algebras. In particular, we show that the Terwilliger algebra for the twisted Grassmann graphs does depend on the base point. More specifically, we show that for the twisted Grassmann graphs there are two vertices x and y such that for the Terwilliger algebra with base point x all the irreducible modules are thin and for the Terwilliger algebra with base point y there exist non-thin irreducible modules. (C) 2008 Elsevier Ltd. All rights reserved.X1111sciescopu
On triangle-free distance-regular graphs with an eigenvalue multiplicity equal to the valency
No full textLet Gamma be a triangle-free distance-regular graph with diameter d >= 3, valency k >= 3 and intersection number a(2) not equal 0. Assume Gamma has an eigenvalue with multiplicity k. We show that Gamma is 1-homogeneous in the sense of Nomura when d = 3 or when d >= 4 and a(4) = 0. In the latter case we prove that r is an antipodal cover of a strongly regular graph, which means that it has diameter 4 or 5. For d = 5 the following infinite family of feasible intersection arrays: {2 mu(2) + mu, 2 mu(2) + mu -1, mu(2), mu,1; 1, mu, mu(2), 2 mu(2) + mu - 1, 2 mu(2) + mu}, mu is an element of N, is known. For mu = 1 the intersection array is uniquely realized by the dodecahedron. For mu = 1 we show that there are no distance-regular graphs with this intersection array. (c) 2007 Elsevier Ltd. All rights reserved.X114sciescopu
Triangle-free distance-regular graphs with an eigenvalue multiplicity equal to their valency and diameter 3
No full textIn this paper, triangle-free distance-regular graphs with diameter 3 and an eigenvalue theta with multiplicity equal to their valency are studied. Let Gamma be such a graph. We first show that theta = -1 if and only if Gamma is antipodal. Then we assume that the graph Gamma is primitive. We show that it is formally self-dual (and hence Q-polynomial and I-homogeneous), all its eigenvalues are integral, and the eigenvalue with multiplicity equal to the valency is either second largest or the smallest. Let x, y is an element of V Gamma be two adjacent vertices, and z is an element of Gamma(2)(x) boolean AND Gamma(2)(y). Then the intersection number tau(2) := vertical bar Gamma(z) boolean AND Gamma(3)(x) boolean AND Gamma(3)(y)vertical bar is independent of the choice of vertices x, y and z. In the case of the coset graph of the doubly truncated binary Golay code, we have b(2) = tau(2). We classify all the graphs with b(2) = tau(2) and establish that the just mentioned graph is the only example. In particular, we rule out an infinite family of otherwise feasible intersection arrays. (c) 2006 Elsevier Ltd. All rights reserved.X112sciescopu
- …
