<p><it>Abstract—</it>An enumeration of star graphs is given which has many useful properties. For example an arbitrary prefix or suffix is connected; indeed the diameter is $<tmath>O(n)</tmath>$. As a consequence, there is an $<tmath>O(n)</tmath>$ interval broadcast algorithm. Prefixes which have $<tmath>t(n-1)!</tmath>$ vertices for some $<tmath>t</tmath>$ are especially well-behaved. The topology of, embeddings in, and algorithms for these graphs are considered, making use of the enumeration.</p><p><it>Index Terms—</it>Interconnection network, star graph, incomplete star graph, interval broadcast, Hamiltonian cycles.</p>