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GeezerVison is showing mathworld.wolfram.com — Intersection Array -- from Wolfram MathWorld

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Intersection Array

The intersection array of G, where G is a distance-regular graph with integers b_i,c_i,i=0,...,d, is the sequence

iota(gamma)={b_0,b_1,...,b_(d-1);c_1,...,c_d}

when, for any two vertices x,y in G at distance i=d(x,y), there are exactly c_i neighbors of y in G_(i-1)(x) and b_i neighbors of y in G_(i+1)(x).

A similar type of intersection array can also be defined for a distance-transitive graph.

A distance-regular graph with global parameters [[c_0,a_0,b_0],[c_1,a_1,b_1],[c_2,a_2,b_2],[c_3,a_3,b_3],[c_4,a_4,b_4]] has intersection array {b_0,b_1,b_2,b_3;c_1,c_2,c_3,c_4}.

See also

Distance-Regular Graph, Distance-Transitive Graph, Global Parameters

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References

Bendito, E.; Carmona, A.; and Encinas, A. M. "Shortest Paths in Distance-Regular Graphs." Europ. J. Combin. 21, 153-166, 2000.Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.Godsil, C. and Royle, G. Algebraic Graph Theory. New York: Springer-Verlag, p. 68, 2001.Koolen, J. H.; Yu, K.; Liang, X.; Choi, H.; and Markowsky, G. "Non-Geometric Distance-Regular Graphs of Diameter at Least 3 with Smallest Eigenvalue at Least -3." 15 Nov 2023. https://arxiv.org/abs/2311.09001.van Dam, E. R. and Haemers, W. H. "Spectral Characterizations of Some Distance-Regular Graphs." J. Algebraic Combin. 15, 189-202, 2003.

Referenced on Wolfram|Alpha

Intersection Array

Cite this as:

Weisstein, Eric W. "Intersection Array." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IntersectionArray.html

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