1. NON-CYCLIC GRAPH ASSOCIATED WITH A GROUP
- Author
-
Alireza Abdollahi and A. Mohammadi Hassanabadi
- Subjects
Combinatorics ,Algebra and Number Theory ,Simple graph ,Locally cyclic group ,Domination analysis ,Solvable group ,Applied Mathematics ,Maximum size ,Graph ,Clique number ,Mathematics ,Vertex (geometry) - Abstract
We associate a graph [Formula: see text] 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 subgraph of Γ. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group G is solvable whenever [Formula: see text] and the equality for a non-solvable group G holds if and only if G/ Cyc (G) ≅ A5 or S5.
- Published
- 2009