Let R be a ring with nonzero identity. The unit graph of R, denoted by G R , has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G R are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G R are given. (These terms are defined in Definitions and Remarks 4.1, 5.1, 5.3, 5.9, and 5.13.)

In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which |G | is prime and G ≤ Z (G) as well as for groups G which |G | is prime and G ∩ Z (G) = 1. This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237-247].

Abstract. Let K be a field and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over the field K. Let G be a forest with p connected components G 1 , . . . , G p and let I = I(G) be its edge ideal in S. Suppose that d i is the diameter of G i , 1 ≤ i ≤ p, and consider d = max {d i | 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max+ p − 1, p is a lower bound for depth(S/I t ). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/I t ). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.