conference graph

{{one source |date=April 2024}}

{{Short description|Special case of a strongly regular graph}}

In the 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.

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,

:$\backslash frac\{-1\; \backslash pm\; \backslash sqrt\; v\}\{2\}\; ,$

each with multiplicity {{nowrap|(v − 1)/2.}}