New constructions of strongly regular Cayley graphs on abelian groups

TL;DR

This paper introduces new methods for constructing strongly regular Cayley graphs on abelian groups, significantly expanding the known families and parameters of such graphs, with implications for combinatorial design theory.

Contribution

The paper extends the theory of building blocks to produce large classes of strongly regular Cayley graphs on abelian groups, generalizing previous constructions and discovering new parameter families.

Findings

Constructed strongly regular Cayley graphs with negative Latin square type parameters.

Decomposed complete graphs into strongly regular Cayley graphs for broader group classes.

Identified new infinite families of Latin square type strongly regular Cayley graphs.

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 $9^iv^4$ 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 $G_i$, $i=1,2,\ldots,s$, 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 $(u^2,c(u+1),- u+c^2+3 c,c^2+ c)$, where $u=q_1^2q_2^2\cdots q_s^2$ 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.