2,983 research outputs found

    On Distance-Regular Graphs with Smallest Eigenvalue at Least βˆ’m-m

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    A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer mβ‰₯2m\geq 2, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least βˆ’m-m, diameter at least three and intersection number c2β‰₯2c_2 \geq 2

    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

    Distance-regular graph with large a1 or c2

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    In this paper, we study distance-regular graphs Ξ“\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 Ξ“\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

    Shilla distance-regular graphs

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    A Shilla distance-regular graph G (say with valency k) is a distance-regular graph with diameter 3 such that its second largest eigenvalue equals to a3. We will show that a3 divides k for a Shilla distance-regular graph G, and for G we define b=b(G):=k/a3. In this paper we will show that there are finitely many Shilla distance-regular graphs G with fixed b(G)>=2. Also, we will classify Shilla distance-regular graphs with b(G)=2 and b(G)=3. Furthermore, we will give a new existence condition for distance-regular graphs, in general.Comment: 14 page

    Distance-regular Cayley graphs with small valency

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    We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most 44, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency 55, and the Cayley graphs among all distance-regular graphs with girth 33 and valency 66 or 77. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table

    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
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