A Classification of the Strongly Regular Generalised Johnson Graphs

Abstract

The generalised Johnson graphs are the graphs J(n, k, m) whose vertices are the k subsets of {1, 2, . . . , n}, with two vertices J1 and J2 joined by an edge if and only if \({{|J_1 \cap J_2| = m}}\) . A graph is called d-regular if every vertex has exactly d edges incident to it. A d-regular graph on v vertices is called a (v, d, a, c)-strongly regular graph if every pair of adjacent vertices have exactly a common neighbours and every pair of non-adjacent vertices have exactly c common neighbours. The triangular graphs J(n, 2, 1), their complements J(n, 2, 0), the sporadic examples J(10, 3, 1) and J(7, 3, 1), as well as the trivially strongly regular graphs J(2k, k, 0) are examples of strongly regular generalised Johnson graphs. In this paper we prove that there are no other strongly regular generalised Johnson graphs.