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For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) θ1 (resp., θD) we show that (θ1+1)(θD+1)?-b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue. 相似文献
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In this paper we will look at the relationship between the intersection number c 2 and the diameter of a distance-regular graph. We also give some tools to show that a distance-regular graph with large c 2 is bipartite, and a tool to show that if k D is too small then the distance-regular graph has to be antipodal. 相似文献
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Benjamin V. C. Collins 《Graphs and Combinatorics》1997,13(1):21-30
Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE i * (x)W≤1 for 0≤i≤d, whereE i * (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent:
- Γis a regular generalized quadrangle.
- Γis thin and c 3=1.
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《Discrete Mathematics》2002,231(1-3):359-360
G. Ehrlich, S. Even, and R.E. Tarjan conjectured that the graph obtained from a complete 3 partite graph K4,4,4 by deleting the edges of four disjoint triangles is not the intersection graph of straight line segments in the plane. We show that it is. 相似文献
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We consider a bipartite distance-regular graph with vertex set , diameter , and valency . For , let denote the set of vertices in that are distance from vertex . We assume there exist scalars , not all zero, such that for all with path-length distances . Fix , and let denote the graph with vertex set and edge set . We show that the adjacency matrix of the local graph has at most four distinct eigenvalues. We are motivated by the fact that our assumption above holds if is -polynomial. 相似文献
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Leonard H. Soicher 《Designs, Codes and Cryptography》2017,84(1-2):101-108
It is known that, up to isomorphism, there is a unique distance-regular graph \(\Delta \) with intersection array \(\{32,27;1,12\}\) [equivalently, \(\Delta \) is the unique strongly regular graph with parameters (105, 32, 4, 12)]. Here we investigate the distance-regular antipodal covers of \(\Delta \). We show that, up to isomorphism, there is just one distance-regular antipodal triple cover of \(\Delta \) (a graph \(\hat{\Delta }\) discovered by the author over 20 years ago), proving that there is a unique distance-regular graph with intersection array \(\{32,27,8,1;1,4,27,32\}\). In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of \(\Delta \), and so no distance-regular graph with intersection array \(\{32,27,6,1;1,6,27,32\}\). We also show there is no distance-regular antipodal 4-cover of \(\Delta \), and so no distance-regular graph with intersection array \(\{32,27,9,1;1,3,27,32\}\), and that there is no distance-regular antipodal 6-cover of \(\Delta \) that is a double cover of \(\hat{\Delta }\). 相似文献
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Let Γ be a distance-regular graph with r = l(1, a1, b1), a1 > 0 and c2r+1 = 1. We show the existence of a collinearity graph of a Moore geometry of diameter r + 1 as a subgraph in Γ. In particular, we show that r = 1. 相似文献
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We give an inequality for the group chromatic number of a graph as an extension of Brooks’ Theorem. Moreover, we obtain a structural theorem for graphs satisfying the equality and discuss applications of the theorem. 相似文献
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Štefko Miklavi? 《Linear algebra and its applications》2009,430(1):251-636
Let Γ denote a distance-regular graph with diameter D?3. Assume Γ has classical parameters (D,b,α,β) with b<-1. Let X denote the vertex set of Γ and let A∈MatX(C) denote the adjacency matrix of Γ. Fix x∈X and let A∗∈MatX(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of MatX(C) generated by A,A∗. We call T the Terwilliger algebra of Γ with respect to x. We show that up to isomorphism there exist exactly two irreducible T-modules with endpoint 1; their dimensions are D and 2D-2. For these T-modules we display a basis consisting of eigenvectors for A∗, and for each basis we give the action of A. 相似文献
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A. K. Dewdney 《Aequationes Mathematicae》1975,13(3):316-316
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A. K. Dewdney 《Aequationes Mathematicae》1976,14(3):357-362
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I. N. Belousov 《Proceedings of the Steklov Institute of Mathematics》2016,295(1):28-37
Possible prime-order automorphisms and their fixed-point subgraphs are found for a hypothetical distance-regular graph with intersection array {39, 36, 1; 1, 2, 39}. It is shownthat graphs with intersection arrays {15, 12, 1; 1, 2, 15}, {35, 32, 1; 1, 2, 35}, and {39, 36, 1; 1, 2, 39} are not vertex-symmetric. 相似文献
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L. Yu. Tsiovkina 《Proceedings of the Steklov Institute of Mathematics》2015,289(1):209-215
Prime divisors of orders of automorphisms and their fixed-point subgraphs are studied for a hypothetical distance-regular graph with intersection array {35, 32, 1; 1, 4, 35}. It is shown that there are no arc-transitive distance-regular graphs with intersection array {35, 32, 1; 1, 4, 35}. 相似文献