# Graphs derived from perfect difference sets

## TL;DR

This paper explores a class of graphs constructed from perfect difference sets, revealing that all known examples correspond to Brown graphs, thus connecting combinatorial design theory with graph theory in the degree-diameter problem.

## Contribution

It establishes that graphs derived from perfect difference sets are isomorphic to Brown graphs, unifying two areas in graph theory and combinatorial design.

## Key findings

- Graphs from perfect difference sets are isomorphic to Brown graphs.
- These graphs have diameter two and optimal order for their degree.
- All known perfect difference sets produce graphs within this family.

## Abstract

We study a family of graphs with diameter two and asymptotically optimal order for their maximum degree, obtained from perfect difference sets. We show that for all known examples of perfect difference sets, the graph we obtain is isomorphic to one of the Brown graphs, a well-known family of graphs in the degree-diameter problem.