There is no (75,32,10,16) strongly regular graph

Abstract

We show that there is no (75,32,10,16) strongly regular graph. The result is obtained by a mix of algebraic and computational approaches. The main idea is to build large enough induced structure and apply the star complement technique. Our result implies that there is no partial geometry pg(4,7,2) and no regular two-graph with 76 vertices, from which it follows that there also exists no strongly regular graph with parameters (76,35,18,14). In order to solve this classification problem we also develop an efficient algorithm for the problem of finding a maximal clique in a graph.