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- Type of Document: Article
- DOI: 10.1142/S0219498815500656
- Publisher: World Scientific Publishing Co. Pte Ltd , 2015
- Abstract:
- Let G be a group. The intersection graph of G, denoted by Γ(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ∩ K ≠= 1. In this paper, we show that the girth of Γ(G) is contained in the set {3,∞ }. We characterize all solvable groups whose intersection graphs are triangle-free. Moreover, we show that if G is finite and Γ (G) is triangle-free, then G is solvable. Also, we prove that if Γ (G) is a triangle-free graph, then it is a disjoint union of some stars. Among other results, we classify all abelian groups whose intersection graphs are complete. Finally, we study the intersection graphs of cyclic groups
- Keywords:
- Group ; Intersection graph
- Source: Journal of Algebra and its Applications ; Volume 14, Issue 5 , June , 2015 ; 02194988 (ISSN)
- URL: http://www.worldscientific.com/doi/10.1142/S0219498815500656