We associate a graph CG to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\Cyc(G) as vertex set, where Cyc(G) = {x ∈ G | 〈x, y〉 is cyclic for all y ∈ G} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete...

let $g$ be a group and $x in g$‎. ‎the cyclicizer of $x$ is defined to be the subset $cyc(x)=lbrace y in g mid langle x‎, ‎yrangle ; {rm is ; cyclic} rbrace$‎. ‎$g$ is said to be a tidy group if $cyc(x)$ is a subgroup for all $x in g$‎. ‎we call $g$ to be a c-tidy group if $cyc(x)$ is a cyclic subgroup for all $x in g setminus k(g)$‎, ‎where $k(g)$ is the intersection of all the cyclicizers in ...