Analytical Enumeration of Circulant Graphs with Prime-Squared Number of Vertices
نویسندگان
چکیده
A method for the analytical enumeration of circulant graphs with p2 vertices, p a prime, is proposed and described in detail. It is based on the use of S-rings and P olya's enumeration technique. Two di erent approaches, \structural" and \multiplier", are developed and compared. As a result we get counting formulae and generating functions (by valency) for non-isomorphic p2-vertex directed and undirected circulant graphs as well as for some natural subclasses of them such as tournaments and self-complementary graphs. These are the rst general enumerative results for circulant graphs for which the so-called Ad am (single-multiplier) isomorphism condition does not hold. Some numerical data and interrelations between formulae are also obtained. The rst expository part of the paper may serve as a self-contained introduction to the use of Schur rings for enumeration.
منابع مشابه
On the Enumeration of Circulant Graphs of Prime Power and Square Free Orders
The aim of this work is twofold: to unify, systematize and extend the known results of the analytical (i.e. formula-wise) counting of directed and undirected circulant graphs with a prime or, more generally, square-free number of vertices; to develop a general combinatorial framework for the counting of nonisomorphic circulant graphs with pk vertices, p odd prime, k 2. The general problem of co...
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