There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some properties of the associated graph. The power graph P(G) of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if y = x m or x = y m for some positive integer m. We also pay attention to the subgraph P

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If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if there is an element in G of order pq. Assume [Formula: see text] with primes p1<p2<⋯<pk and natural numbers αi. For p∈π(G), let the degree of p be deg (p)=|{q∈π(G)|q~p}|, and D(G):=( deg (p1), deg (p2),…, deg (pk)). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups Ap with p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC (G)={29.39.5.7, 13}, then G≅S6(3) or O7(3), and finally, we show that if G is a finite group such that |G|=29.39.5.7.13 and D(G)=(3,2,2,1,0), then G≅S6(3) or O7(3).

Let G be a finite group. We define the noncommuting graph ∇(G) as follows: the vertex set of ∇(G) is G \ Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. We study some properties of ∇(G) and prove that, for many groups G, if H is a group with ∇(G) isomorphic to ∇(H) then |G| = |H|.

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