Abstract. The commuting graph of a non-abelian group is a simple graph in which the vertices are the non-central elements of the group, and two distinct vertices are adjacent if and only if they commute. In this paper, we determine (up to isomorphism) all finite non-abelian groups whose commuting graphs are acyclic, planar or toroidal. We also derive explicit formulas for the genus of the commuting graphs of some well-known class of finite non-abelian groups, and show that, every collection of isomorphism classes of finite non-abelian groups whose commuting graphs have the same genus is finite.Mathematics Subject Classification (2010): 20D60, 05C25
Abstract. Let G be a group and nil(G) = {x ∈ G | x, y is nilpotent for all y ∈ G}. Associate a graph R G (called the non-nilpotent graph of G) with G as follows: Take G \ nil(G) as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of R G . We conjecture that the domination number of the non-nilpotent graph of every finite non-abelian simple group is 2. We also conjecture that if G and H are two non-nilpotent finite groups such thatAmong other results, we show that the nonnilpotent graph of D 10 is double-toroidal.
Mathematics Subject Classification (2010): 20D60
Let G be a finite non-solvable group with solvable radical Sol(G). The solvable graph Γs(G) of G is a graph with vertex set G \ Sol(G) and two distinct vertices u and v are adjacent if and only if u, v is solvable. We show that Γs(G) is not a star graph, a tree, an n-partite graph for any positive integer n ≥ 2 and not a regular graph for any non-solvable finite group G. We compute the girth of Γs(G) and derive a lower bound of the clique number of Γs(G). We prove the nonexistence of finite non-solvable groups whose solvable graphs are planar, toroidal, double-toroidal, triple-toroidal or projective. We conclude the paper by obtaining a relation between Γs(G) and the solvability degree of G.
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