Geodesic
Analytical Enumeration of Circulant Graphs with Prime-Squared Number of Vertices.
Séminaire lotharingien de combinatoire, Tome 36 (1996)

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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 Pólya's enumeration technique. Two different 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 subclasses of them such as tournaments and self-complementary graphs. These are the first general enumerative results for circulant graphs for which the so-called Ádám (single-multiplier) isomorphism condition does not hold. Some numerical data and interrelations between formulae are also obtained. The first expository part of the paper may serve as a self-contained introduction to the use of Schur rings for enumeration. 

@article{SLC_1996_36_a3,
     author = {Mikhail Klin and Valery Liskovets and Reinhard P\"oschel},
     title = {Analytical {Enumeration} of {Circulant} {Graphs} with {Prime-Squared} {Number} of {Vertices.}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {36},
     year = {1996},
     url = {http://geodesic.mathdoc.fr/item/SLC_1996_36_a3/}
}
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AU  - Valery Liskovets
AU  - Reinhard Pöschel
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%U http://geodesic.mathdoc.fr/item/SLC_1996_36_a3/
%F SLC_1996_36_a3
Mikhail Klin; Valery Liskovets; Reinhard Pöschel. Analytical Enumeration of Circulant Graphs with Prime-Squared Number of Vertices.. Séminaire lotharingien de combinatoire, Tome 36 (1996). http://geodesic.mathdoc.fr/item/SLC_1996_36_a3/