Third Zagreb Index of Graphs with Added Edges

Edge deletion and addition to a graph is an important combinatorial method in Graph Theory which enables one to calculate some properties of a graph by means of similar and usually simpler graphs. In this paper, as a sequel to recent papers on edge deletion and addition, we consider the hange in the third Zagreb index of a simple graph G when an arbitrary edge is added. The effect of adding any kind of edge to a graph is shown to be an integer congruent to 2 modulo 6. This result an cbe used to calculate the third Zagreb index of larger graphs in terms of the Zagreb indices of smaller graphs. As some examples, some inequalities for the hange of Zagreb indices of path, cycle, star, complete, complete bipartite and tadpole graphs are given.