Graphs of order two less than the Moore bound

The problem of determining the largest order nd,knd,k of a graph of maximum degree at most d and diameter at most k is well known as the degree/diameter problem  . It is known that nd,k⩽Md,knd,k⩽Md,k where Md,kMd,k is the Moore bound. For d=4d=4, the current best upper bound for n4,kn4,k is M4,k-1M4,k-1. In this paper we study properties of graphs of order Md,k-2Md,k-2 and we give a new upper bound for n4,kn4,k for k⩾3k⩾3.