Almost-Ramanujan graphs and prime gaps

On Construction of Almost-Ramanujan Graphs

Reingold et al. introduced the notion zig-zag product on two different graphs, and presented a fully explicit construction of dregular expanders with the second largest eigenvalue O(d−1/3). In the same paper, they ask whether or not the similar technique can be used to construct expanders with the second largest eigenvalue O(d−1/2). Such graphs are called Ramanujan graphs. Recently, zig-zag pro...

Integral Circulant Ramanujan Graphs of Prime Power Order

A connected ρ-regular graph G has largest eigenvalue ρ in modulus. G is called Ramanujan if it has at least 3 vertices and the second largest modulus of its eigenvalues is at most 2 √ ρ− 1. In 2010 Droll classified all Ramanujan unitary Cayley graphs. These graphs of type ICG(n, {1}) form a subset of the class of integral circulant graphs ICG(n,D), which can be characterised by their order n an...

On generalisations of almost prime and weakly prime ideals

Let $R$ be a commutative ring with identity‎. ‎A proper ideal $P$ of $R$ is a $(n-1,n)$-$Phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin R$‎, ‎$a_1cdots a_nin Pbackslash P^m$ ($a_1cdots a_nin Pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin P$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎In this paper several results concerning $(n-1,n)$-$Phi_m$-prime and $(n-1,n)$-...

Ramanujan graphs

Ramanujan Graphs

In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication network theory (see for example [2]). Second, from a more aesthetic viewpoint, they fuse diverse branches of pure mathematics, namely, number theory, representation theory and algebraic geometry. The...