On the Nonexistence of Strongly Regular Graphs with Parameters (486, 165, 36, 66)

Abstract

We prove that a strongly regular graph with parameters (486, 165, 36, 66) does not exist. Since the parameters indicated are parameters of a pseudogeometric graph for pG ₂(5, 32), we conclude that the partial geometries pG ₂(5, 32) and pG ₂(32, 5) do not exist. Finally, a neighborhood of an arbitrary vertex of a pseudogeometric graph for pG ₃(6, 80) is a pseudogeometric graph for pG ₂(5, 32) and, therefore, a pseudogeometric graph for the partial geometry pG ₃(6, 80) [i.e., a strongly regular graph with parameters (1127, 486, 165, 243)] does not exist.