On a family of strongly regular graphs with \lambda=1

TL;DR

This paper classifies all strongly regular graphs with specific parameters, identifying three known graphs: the lattice graph, the Brouwer-Haemers graph, and the Games graph, providing a complete description of this family.

Contribution

It provides a complete classification of strongly regular graphs with parameters ((n^2+3n-1)^2,n^2(n+3),1,n(n+1)), identifying all such graphs explicitly.

Findings

Identified three specific strongly regular graphs with given parameters.

Provided a complete description of the family of graphs with these parameters.

Confirmed the uniqueness of these graphs within the specified parameter set.

Abstract

In this paper, we give a complete description of strongly regular graphs with parameters ((n^2+3n-1)^2,n^2(n+3),1,n(n+1)). All possible such graphs are: the lattice graph $L_{3,3}$ with parameters (9,4,1,2), the Brouwer-Haemers graph with parameters (81,20,1,6), and the Games graph with parameters (729,112,1,20).