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# Conference Graph

A strongly regular graph with parameters has graph eigenvalues , , and , where

 (1) (2)

where

 (3)

(Godsil and Royle 2001, pp. 221-222). In the case of and distinct, call their multiplicities in the graph spectrum and . Then a graph with is called a conference graph. All Paley graphs are conference graphs.

A strongly regular graph is either a conference graph, has and integers and a perfect square (correcting a typo in Godsil and Royle 2001, p. 222), or both of the above (Godsil and Royle 2001, p. 222). Paley graphs with a square number (including the (2,1)-generalized quadrangle, which is isomorphic to the 9-Paley graph) satisfy both conditions.

In the special case that is a strongly regular graph with vertices where is prime, is a conference graph (Godsil and Royle 2001, p. 222).

The following table summarizes some conference graphs.

 graph characteristic polynomial 5 5-cycle graph 9 -generalized quadrangle 13 13-Paley 17 17-Paley 25 25-Paley 25 25-Paley 25 25-Paulus graph 1-14 29 29-Paley 37 37-Paley 41 41-Paley 49 49-Paley 53 53-Paley 61 61-Paley 73 73-Paley 81 81-Paley 89 89-Paley 97 97-Paley 101 101-Paley 109 109-Paley 113 113-Paley 121 121-Paley 125 125-Paley 137 137-Paley 149 149-Paley 157 157-Paley 169 169-Paley

C-Matrix, Graph Eigenvalue, Graph Spectrum, Paley Graph, Paulus Graphs, Strongly Regular Graph

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## References

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. 222, 2001.

Conference Graph

## Cite this as:

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