In graph theory, a strongly regular graph is defined as follows. Let G = be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers λ and μ such that:
The four parameters in an srg are not independent and must obey the following relation: The above relation can be derived very easily through a counting argument as follows:
Imagine the vertices of the graph to lie in three levels. Pick any vertex as the root, in Level 0. Then its k neighbors lie in Level 1, and all other vertices lie in Level 2.
Vertices in Level 1 are directly connected to the root, hence they must have λ other neighbors in common with the root, and these common neighbors must also be in Level 1. Since each vertex has degree k, there are edges remaining for each Level 1 node to connect to nodes in Level 2. Therefore, there are edges between Level 1 and Level 2.
Vertices in Level 2 are not directly connected to the root, hence they must have μ common neighbors with the root, and these common neighbors must all be in Level 1. There are vertices in Level 2, and each is connected to μ nodes in Level 1. Therefore the number of edges between Level 1 and Level 2 is.
Equating the two expressions for the edges between Level 1 and Level 2, the relation follows.
Adjacency Matrix
Let I denote the identity matrix and let J denote the matrix of ones, both matrices of order v. The adjacency matrixA of a strongly regular graph satisfies two equations. First: which is a trivial restatement of the regularity requirement. This shows that k is an eigenvalue of the adjacency matrix with the all-ones eigenvector. Second is a quadratic equation, which expresses strong regularity. The ij-th element of the left hand side gives the number of two-step paths from i to j. The first term of the RHS gives the number of self-paths from i to i, namely k edges out and back in. The second term gives the number of two-step paths when i and j are directly connected. The third term gives the corresponding value when i and j are not connected. Since the three cases are mutually exclusive andcollectively exhaustive, the simple additive equality follows. Conversely, a graph whose adjacency matrix satisfies both of the above conditions and which is not a complete or null graph is a strongly regular graph.
Eigenvalues
The adjacency matrix of the graph has exactly three eigenvalues:
As the multiplicities must be integers, their expressions provide further constraints on the values of v, k, μ, and λ, related to the so-called Krein conditions. Strongly regular graphs for which are called conference graphs because of their connection with symmetric conference matrices. Their parameters reduce to Strongly regular graphs for which have integer eigenvalues with unequal multiplicities. Conversely, a connected regular graph with only three eigenvalues is strongly regular.
The n × n square rook's graph, i.e., the line graph of a balanced complete bipartite graphKn,n, is an srg. The parameters for n=4 coincide with those of the Shrikhande graph, but the two graphs are not isomorphic.
The Chang graphs are srg, the same as the line graph of K8, but these four graphs are not isomorphic.
The line graph of a generalized quadrangle GQ is an srg. In fact every generalized quadrangle of order gives a strongly regular graph in this way: to wit, an srg, s.
A strongly regular graph is called primitive if both the graph and its complement are connected. All the above graphs are primitive, as otherwise μ=0 or λ=k. Conway's 99-graph problem asks for the construction of an srg. It is unknown whether a graph with these parameters exists, and John Horton Conway has offered a $1000 prize for the solution to this problem.
Moore graphs
The strongly regular graphs with λ = 0 are triangle free. Apart from the complete graphs on less than 3 vertices and all complete bipartite graphs the seven listed above are the only known ones. Strongly regular graphs with λ = 0 and μ = 1 are Moore graphs with girth 5. Again the three graphs given above, with parameters, and, are the only known ones. The only other possible set of parameters yielding a Moore graph is ; it is unknown if such a graph exists, and if so, whether or not it is unique.
