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

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.