1,533 research outputs found

    An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

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    For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+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.Comment: 15 pages, this is submitted to Linear Algebra and Applications

    Taut distance-regular graphs and the subconstituent algebra

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    We consider a bipartite distance-regular graph GG with diameter DD at least 4 and valency kk at least 3. We obtain upper and lower bounds for the local eigenvalues of GG in terms of the intersection numbers of GG and the eigenvalues of GG. Fix a vertex of GG and let TT denote the corresponding subconstituent algebra. We give a detailed description of those thin irreducible TT-modules that have endpoint 2 and dimension D−3D-3. In an earlier paper the first author defined what it means for GG to be taut. We obtain three characterizations of the taut condition, each of which involves the local eigenvalues or the thin irreducible TT-modules mentioned above.Comment: 29 page

    Distance-regular graph with large a1 or c2

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    In this paper, we study distance-regular graphs Γ\Gamma 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\Gamma. 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.Comment: We submited this manuscript to JCT

    Distance-regular graphs

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    This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

    Eigenvalue interlacing and weight parameters of graphs

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    Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some "weights" (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights "regularize" the graph, and hence allow us to define a kind of regular partition, called "pseudo-regular," intended for general graphs. Here we show how to use interlacing for proving results about some weight parameters and pseudo-regular partitions of a graph. For instance, generalizing a well-known result of Lov\'asz, it is shown that the weight Shannon capacity Θ∗\Theta^* of a connected graph \G, with nn vertices and (adjacency matrix) eigenvalues λ1>λ2≥.˙.≥λn\lambda_1>\lambda_2\ge\...\ge \lambda_n, satisfies \Theta\le \Theta^* \le \frac{\|\vecnu\|^2}{1-\frac{\lambda_1}{\lambda_n}} where Θ\Theta is the (standard) Shannon capacity and \vecnu is the positive eigenvector normalized to have smallest entry 1. In the special case of regular graphs, the results obtained have some interesting corollaries, such as an upper bound for some of the multiplicities of the eigenvalues of a distance-regular graph. Finally, some results involving the Laplacian spectrum are derived. spectrum are derived
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