We investigate the group irregularity strength (sg(G)) of graphs, that is, we find the minimum value of s such that for any Abelian group G of order s, there exists a function f : E(G) → G such that the sums of edge labels at every vertex are distinct. We prove that for any connected graph G of order at least 3, sg(G) = n if n = 4k + 2 and sg(G) ≤ n + 1 otherwise, except the case of an infinite...

The irregularity of a simple undirected graph G was defined by Albertson [5] as irr(G) = ∑ uv∈E(G) |dG(u)− dG(v)|, where dG(u) denotes the degree of a vertex u ∈ V (G). In this paper we consider the irregularity of graphs under several graph operations including join, Cartesian product, direct product, strong product, corona product, lexicographic product, disjunction and symmetric difference. ...

the total version of geometric–arithmetic index of graphs is introduced based on the endvertexdegrees of edges of their total graphs. in this paper, beside of computing the total gaindex for some graphs, its some properties especially lower and upper bounds are obtained.