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In the mathematical area of graph theory, a 'conference graph' is a strongly regular graph with parameters ''v'', ''k'' = (''v''−1)/2, λ = (''v''−5)/4, and μ = (''v''−1)/4. It is the graph associated with a symmetric conference matrix, and consequently its order ''v'' must be 1 (modulo 4) and a sum of two squares.
Conference graphs are known to exist for all small values of ''v'' allowed by the restrictions, e.g., ''v'' = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). However, there are many values of ''v'' that are allowed, for which the existence of a conference graph is unknown.
The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs. If the graph is connected, the eigenvalues are ''k'' with multiplicity 1, and two other eigenvalues,
:
each with multiplicity (''v''−1)/2.
Brouwer, A.E., Cohen, A.M., and Neumaier, A. (1989), Distance Regular Graphs. Berlin, New York: Springer-Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5
Conference graphs are known to exist for all small values of ''v'' allowed by the restrictions, e.g., ''v'' = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). However, there are many values of ''v'' that are allowed, for which the existence of a conference graph is unknown.
The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs. If the graph is connected, the eigenvalues are ''k'' with multiplicity 1, and two other eigenvalues,
:
each with multiplicity (''v''−1)/2.
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| References |
References
Brouwer, A.E., Cohen, A.M., and Neumaier, A. (1989), Distance Regular Graphs. Berlin, New York: Springer-Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5
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