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 abelian non p-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 $G_i$, $i=1,2,\dots ,s$, of order $q_i^4$ such that $2m|q_i+1$, there exists a decomposition of the complete graph on the abelian group $G=G_1\times G_2\times \cdots \times G_s$ by strongly regular Cayley graphs with negative Latin square type parameters $(u^2,c(u+1),-u+c^2+3c,c^2+c)$, where $u=q_1^2{q}_2^2\cdots q_s^2$ 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.

Davis 和 Jedwab (1997) 建立了一个伟大的构造理论，统一了许多先前已知的差分集、相对差分集和可分差分集的构造。他们引入了积木的概念，在理论中发挥了重要作用。另一方面，Polhill (2010) 给出了基于特殊的积木系统的 Paley 型偏差分集（会议图）的构造，称为覆盖扩展构建集，并证明了在阿贝尔有序群$9^{\mathrm{一世}}{v}^4$ 对于任何奇数正整数 $v>1$ 和任何 $\mathrm{一世}=0,1$. 他的结果涵盖了所有存在 Paley 型偏差分集的阿贝尔非p 群阶。在本文中，我们通过扩展积木理论，给出了阿贝尔群上强正则凯莱图的新构造。这些构造是 Polhill 构造的大概括。特别地，我们证明对于正整数m和基本阿贝尔群$G_{\mathrm{一世}}$, $\mathrm{一世}=1,2,\dots ,秒$, 按顺序 $q_{\mathrm{一世}}^4$ 以至于 $2米|q_{\mathrm{一世}}+1$，在阿贝尔群上存在完全图的分解 $G=G_1\times G_2\times \cdots \times G_{秒}$ 通过具有负拉丁方型参数的强正则 Cayley 图 $({你}^2,C(你+1),-你+C^2+3C,C^2+C)$， 在哪里 $你=q_1^2{q}_2^2\cdots q_{秒}^2$ 和 $C=(你-1)/米$. 这种强规则的分解以前只有在$米=2$或G是p 基团。此外，我们通过拉丁方型强正则凯莱图找到了一个新的无限分解族。因此，我们获得了许多具有新参数的强正则图。