Abstract
It is known that a strongly regular semi-Cayley graph (with respect to a group G) corresponds to a triple of subsets (C, D, D′) of G. Such a triple (C, D, D′) is called a partial difference triple. First, we study the case when D ∪ D′ is contained in a proper normal subgroup of G. We basically determine all possible partial difference triples in this case. In fact, when \vert G\vert \neq 8 nor 25, all partial difference triples come from a certain family of partial difference triples. Second, we investigate partial difference triples over cyclic group. We find a few nontrivial examples of strongly regular semi-Cayley graphs when vGv is even. This gives a negative answer to a problem raised by de Resmini and Jungnickel. Furthermore, we determine all possible parameters when G is cyclic. Last, as an application of the theory of partial difference triples, we prove some results concerned with strongly regular Cayley graphs.
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