New constructions of strongly regular Cayley graphs on abelian groups

Abstract

Davis and Jedwab (1997) established a great construction theory unifying many previously known constructions of difference sets, relative difference sets and divisible difference sets. They introduced the concept of building blocks, which played an important role in the theory. On the other hand, Polhill (2010) gave a construction of Paley type partial difference sets (conference graphs) based on a special system of building blocks, called a covering extended building set, and proved that there exists a Paley type partial difference set in an abelian group of order 9 i v 4 for any odd positive integer v > 1 and any i = 0, 1. His result covers all orders of nonelementary abelian groups in which Paley type partial difference sets exist. In this paper, we give new constructions of strongly regular Cayley graphs on abelian groups by extending the theory of building blocks. The constructions are large generalizations of Polhill's construction. In particular, we show that for a positive integer m and elementary abelian groups G i , i = 1, 2, . . . , s, of order q 4 i such that 2m | q i + 1, there exists a decomposition of the complete graph on the abelian group G = G 1 ×G 2 ו • •×G s by strongly regular Cayley graphs with negative Latin square type parameters (u 2 , c(u + 1), −u + c 2 + 3c, c 2 + c), where u = q 2 1 q 2 2 • • • q 2 s and c = (u − 1)/m. Such strongly regular decompositions were previously known only when m = 2 or G is a p-group. Moreover, we find one more new infinite family of decompositions of the complete graphs by Latin square type strongly regular Cayley graphs. Thus, we obtain many strongly regular graphs with new parameters.

Packings of partial difference sets

“…A final construction class uses cyclotomy [46] and character sums [64,70] over Galois domains (direct products of finite fields) to produce partial difference sets in the direct product of elementary abelian groups.…”

“…The construction of partial difference sets is therefore of great interest. We refer to [13,60] for excellent surveys of partial difference sets and equivalent structures, and to [1,2,4,5,8,9,11,12,14,15,17,18,19,20,21,22,24,25,26,27,28,29,30,31,32,37,38,42,43,44,45,46,47,48,49,50,51,52,54,58,59,61,62,63,64,65,67,68,69,…”