Independence and domination number of order two element graph over a group

Let G be a group. Then the order two element graph of G is the simple undirected graph whose vertex set consists of all elements of G and two distinct vertices u,v are adjacent if and only if either uv or vu is an element of {t(2) : t is an element of G } boolean OR {a is an element of G : a(2)= e}\{e}, where e is the identity element of G. This graph is denoted by S-2(G). In this paper, we first establish some beautiful results on Merrifield-Simmons index of order two element graph over a group. After that we characterize the product of two order two element graph over two groups. Moreover we study some important results on domination number of order two element graph over a group.