A Generalization of Brown's Construction for the Degree/Diameter Problem

TL;DR

This paper generalizes Brown's construction to improve lower bounds on the maximum size of degree-2 graphs with diameter 2, specifically for degrees 306 and 307, achieving larger graphs than previously known.

Contribution

It extends Brown's construction method, providing new larger graphs for degree 306 and 307 with diameter 2, thus improving known lower bounds.

Findings

Constructed a (306,2)-graph with 88723 vertices

Constructed a (307,2)-graph with 88724 vertices

Improved lower bounds for degree/diameter problem at these degrees

Abstract

The degree/diameter problem is the problem of finding the largest possible number of vertices $n_{\Delta,D}$ in a graph of given degree $\Delta$ and diameter $D$. We consider the problem for the case of diameter $D=2$. William G Brown gave a lower bound of the order of $(\Delta,2)$-graph. In this paper, we give a generalization of his construction and improve the lower bounds for the case of $\Delta=306$ and $\Delta=307$. One is $(306,2)$-graph with $88723$ vertices, the other is $(307,2)$-graph with $88724$ vertices.