l Geometric Distance-Regular Graphs with Smallest Eigenvalue −3
Speaker
l Bang Sejeong (Pusan National University)
Date
l 2009-08-21
Link
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DownLoad
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Etc
l 2009 Combinatorics Workshop
A geometric distance-regular graph is the point graph of a linear space in which the set of lines
are a set of Delsarte cliques. Geometric strongly regular graphs were introduced by R.C.Bose
([1]), and C.Godsil ([2]) generalized it to distance-regular graphs.
Definition: ([2]) A distance-regular graph Γ with valency k ≥ 3, diameter D ≥ 2 and smallest
eigenvalue θD is called geometric if there exists a set of cliques C satisfying the following:
(i) Each edge lies in exactly one clique in C
(ii) Each clique in C has size 1 −k/\theta_D.
Examples of geometric distance-regular graphs are the Hamming graphs (and more general the
regular 2D-gons), the Johnson graphs, the Grassmann graphs and the bilinear forms graphs.
In this talk, we classify geometric distance-regular graphs with smallest eigenvalue −3 and
intersection number c2 ≥ 2.
References
[1] R. C. Bose, Strongly regular graphs, partial geometries and partially balanced designs,
Pacific J. Math. 13 389-419,1963.
[2] C. D. Godsil, Geometric distance-regular covers, New Zealand J. Math. 22 31–8, 1993.
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