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

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