We present 15 new partial difference sets over 4 non-abelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was an open question. Our method is based on combination of Galois correspondence between permutation groups and association schemes, classical Seidel's switching of edges and essential use of computer algebra packages. As a by-product, a few new amorphic association schemes with 3 classes on 100 points are discovered.

A method for the analytical enumeration of circulant graphs with p 2 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 different approaches, "structural " and "multiplier", are developed and compared. As a result we get counting formulae and generating functions (by valency) for non-isomorphic p 2-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 first 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 first expository part of the paper may serve as a self-contained introduction to the use of Schur rings for enumeration.

We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.

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