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On Some Expander Graphs and Algebraic Cayley Graphs

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 $k$.