In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

area of

graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, a conference graph is a

strongly regular graph In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have comm ...

with parameters ''v'', and It is the graph associated with a symmetric conference matrix, and consequently its order ''v'' must be 1 (

modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ...

4) and a

sum of two squares In number theory, the sum of two squares theorem relates the prime decomposition of any integer to whether it can be written as a sum of two squares, such that for some integers , . :''An integer greater than one can be written as a sum of two s ...

. 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 In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, ...

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, :

\frac ,

each with multiplicity

References

Brouwer, A.E., Cohen, A.M., and Neumaier, A. (1989), Distance Regular Graphs. Berlin, New York: Springer-Verlag. , {{ISBN, 0-387-50619-5 Algebraic graph theory Graph families Strongly regular graphs