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# Shrikhande Graph

The Shrikhande graph is a strongly regular graph on 16 nodes. It is cospectral with the rook graph , so neither of the two is determined by spectrum.

The Shrikhande graph is the smallest distance-regular graph that is not distance-transitive (Brouwer et al. 1989, p. 136). It has intersection array .

The Shrikhande graph is implemented in the Wolfram Language as GraphData["ShrikhandeGraph"].

The Shrikhande graph has two generalized LCF notations of order 8, eleven of order 4, 53 of order 2, and 2900 of order 1. The graphs with LCF notations of orders four and eight are illustrated above.

The Shrikhande graph appears on the cover of the book Combinatorial Matrix Theory by Brualdi and Ryser (1991); illustrated above.

The plots above show the adjacency, incidence, and graph distance matrices for the Shrikhande graph.

It is an integral graph with graph spectrum .

The bipartite double graph of the Shrikhande graph is the Kummer graph.

The following table summarizes some properties of the Shrikhande graph.

 property value automorphism group order 192 characteristic polynomial chromatic number 4 chromatic polynomial ? claw-free no clique number 3 graph complement name ? determined by spectrum no diameter 2 distance-regular graph yes dual graph name Dyck graph edge chromatic number 6 edge connectivity 6 edge count 48 edge transitive yes Eulerian yes girth 3 Hamiltonian yes Hamiltonian cycle count 562464 Hamiltonian path count ? integral graph yes independence number 4 line graph no line graph name ? perfect matching graph no planar no polyhedral graph no radius 2 regular yes square-free no strongly regular parameters symmetric yes traceable yes triangle-free no vertex connectivity 6 vertex count 16 vertex transitive yes

Cospectral Graphs, Determined by Spectrum, Doob Graph, Integral Graph, Strongly Regular Graph, Triangular Graph

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## References

Brouwer, A. E. "Shrikhande Graph." http://www.win.tue.nl/~aeb/drg/graphs/Shrikhande.html.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, pp. 104-105 and 136, 1989.Brouwer, A. E. and van Lint, J. H. "Strongly Regular Graphs and Partial Geometries." In Enumeration and Design: Papers from the conference on combinatorics held at the University of Waterloo, Waterloo, Ont., June 14-July 2, 1982 (Ed. D. M. Jackson and S. A. Vanstone). Toronto, Canada: Academic Press, pp. 85-122, 1984.Brualdi, R. and Ryser, H. J. Combinatorial Matrix Theory. New York: Cambridge University Press, p. 153, 1991.DistanceRegular.org. "Shrikhande Graph." http://www.distanceregular.org/graphs/shrikhande.html.Shrikhande, S.-C. S. "The Uniqueness of the Association Scheme." Ann. Math. Stat. 30, 781-798, 1959.van Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their Spectrum?" Lin. Algebra Appl. 373, 139-162, 2003.

## Cite this as:

Weisstein, Eric W. "Shrikhande Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ShrikhandeGraph.html