On Some Expander Graphs and Algebraic Cayley Graphs

TL;DR

This paper constructs new families of Ramanujan graphs, including infinite classes of k-regular graphs, using algebraic methods involving difference sets, addressing open questions about their existence for all positive integers k.

Contribution

It introduces novel algebraic constructions of Ramanujan graphs using difference sets, providing infinite families for various degrees and solving longstanding existence questions.

Findings

Constructed infinite families of Ramanujan graphs for specific degrees.

Proved existence of k-regular Ramanujan graphs for all even k>4 under certain conditions.

Established existence of Ramanujan graphs for all odd degrees m.

Abstract

Expander graphs have many interesting applications in communication networks and other areas, and thus these graphs have been extensively studied in theoretic computer sciences and in applied mathematics. In this paper, we use reversible difference sets and generalized difference sets to construct more expander graphs, some of them are Ramanujan graphs. Three classes of elementary constructions of infinite families of Ramanujan graphs are provided. It is proved that for every even integer $k>4$, if $2(k+2)=rs$ for two even numbers $r$ and $s$ with $s\geq 4$ and $2s>r\geq s$, or $r\geq 4$ and $2r>s\geq r$, then there exists an $k$-regular Ramanujan graph. As a consequence, there exists an $k$-regular Ramanujan graph with $k=2t^2-2$ for every integer $t>2$. It is also proved that for every odd integer $m$, there is an $(2^{2m-2}+2^{m-1})$-regular Ramanujan graph. These results partially solved the long hanging open question for the existence of $k$-regular Ramanujan graphs for every positive integer $k$.