Fenske equation: Difference between revisions

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In the [[mathematics|mathematical]] area of [[graph theory]], a '''conference graph''' is a [[strongly regular graph]] with parameters ''v'', {{nowrap|1=''k'' = (''v'' − 1)/2,}} {{nowrap|1=λ = (''v'' − 5)/4,}} and {{nowrap|1=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.
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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 graph]]s) 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,
:<math>\frac{-1 \pm \sqrt v}{2} , </math>  
each with multiplicity {{nowrap|(''v'' &minus; 1)/2.}}
 
==References==
 
[[Andries Brouwer|Brouwer, A.E.]], Cohen, A.M., and Neumaier, A. (1989), <cite>Distance Regular Graphs</cite>.  Berlin, New York: Springer-Verlag.  ISBN 3-540-50619-5, ISBN 0-387-50619-5
 
[[Category:Algebraic graph theory]]
[[Category:Graph families]]

Latest revision as of 02:49, 17 August 2014

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