Comparison of Two Techniques for Proving Nonexistence of Strongly Regular Graphs

We show that the method of counting closed walks in strongly regular graphs rules out no parameter sets other than those ruled out by the method of counting eigenvalue multiplicities. Following Bose [2], a strongly regular graph with parameters n, k, λ, μ means an undirected graph G such that • G has n vertices, • G is regular of degree k, • every two adjacent vertices of G have precisely λ common neighbours, • every two nonadjacent vertices of G have precisely μ common neighbours. Complete graphs have these four properties (with k = n − 1, λ = n − 2, and any μ) and so have their complements (with k = n− 1, any λ, and μ = n− 2). Let us follow the convention of excluding these trivial examples from the class of strongly regular graphs: let us assume that 0 < k < n− 1. (1) If there exists a strongly regular graph with parameters n, k, λ, μ, then (n− 1− k)μ = k(k − 1− λ). (2) (This identity follows directly from counting in two different ways all sequences w0, w1, w2 of vertices w0, w1, w2 such that w0 is prescribed, w0, w1 are nonadjacent, w1, w2 are adjacent, and w0, w2 are nonadjacent: choosing w2 first and w1 second gives the left-hand side; choosing w1 first and w2 second gives the righthand side). Another widely known condition that is necessary for the existence of a strongly regular graph with parameters n, k, λ, μ goes as follows: ∗Department of Computer Science and Software Engineering, Concordia University, Montréal, Québec, Canada