The Enumeration of Homeomorphically Irreducible Star Graphs

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Abstract
In expressing properties of interacting systems in the form of series, in many cases only summation over star graphs is involved. The identification and classification of such graphs is simplified by reducing them to homeomorphically irreducible stars. These graphs can be regarded as being all the possible different topological types of star. A method is described which has been used to produce all the different homeomorphically irreducible stars which have cyclomatic numbers ≤ 5. In particular, it has been established that there are 118 such graphs with cyclomatic number 5, 111 of which are planar. Diagrams of these graphs are appended, and a table of their k weights, which are needed for obtaining series for percolation processes, is also given. An extension of the method has been used to count the numbers of homeomorphically irreducible stars containing up to 8 vertices and 12 edges, a table of which is given.
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