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Generalized Moore graphs are regular graphs that satisfy an additional distance condition, namely, that there be the maximum number of vertices as close as possible to any particular vertex, when that vertex is considered as root vertex. These graphs form a useful model for the study of various theoretical properties of computer communications networks. In particular, they lend themselves to a discussion of lower bounds for network cost, delay, reliability, and vulnerability. A considerable number of papers have already been published concerning the existence and properties of generalized Moore graphs of valence three, and some initial studies have discussed generalized Moore graphs of valence four, when the number of vertices is less than fourteen. This paper continues the previous studies for those cases when the graph contains a number of vertices that is between fourteen and twenty. In the case of valence three, the graph with a complete second level exists; it is just the Petersen graph. The situation is quite different for valence four; not only does the graph with a complete second level not exist, but the graphs in its immediate “neighbourhood” also fail to exist.