The commuting graph ∆(G) of a group G is the graph whose vertex set is the group and two distinct elements x and y being adjacent if and only if xy = yx. In this paper the automorphism group of this graph is investigated. We observe that Aut(∆(G)) is a non-abelian group such that its order is not prime power and square-free.

Let D be a division ring, n 2 a natural number, and C ⊆ Mn(D). Two matrices A and B are called C−commuting if there is C ∈ C that AB−BA = C. In this paper the C−commuting graph of Mn(D) is defined and denoted by ΓC(Mn(D)). Conditions are given that guarantee that the C−commuting graph is connected.

Let D be a division ring, n 2 a natural number, and C ⊆ Mn(D). Two matrices A and B are called C−commuting if there is C ∈ C that AB−BA = C. In this paper the C−commuting graph of Mn(D) is defined and denoted by ΓC(Mn(D)). Conditions are given that guarantee that the C−commuting graph is connected.

Let G be a non-abelian group and Z(G) be its center. The non-commuting graph AG of G is the graph whose vertex set is G\Z(G) and two vertices are joined by an edge if they do not commute. Let SL(2, q) be the special linear group of degree 2 over the finite field of order q. In this paper we prove that if G is a group such that AG ∼= ASL(2,q) for some prime power q ≥ 2, then G ∼= SL(2, q). MSC 2...

The author studies trends in commuting to the city of Zagreb in Croatia, Yugoslavia. Commuter surveys and bivariate analysis are used to describe the socioeconomic level, occupational status, transportation method, and place of residence of rural and urban residents who commute to the city. (SUMMARY IN ENG AND RUS)

The purpose of this note is to define a graph whose vertex set finite group G $G$ , edge contained in that the commuting and contains enhanced power . We call deep Two elements are joined if only their inverse images every central extension commute. give conditions for be equal either graph, show automorphisms act as graph.