Generalized four new sums of graphs and their Zagreb indices

The first Zagreb index of a graph G is the sum of squares of the degrees of the vertices of G. In this paper, we introduce generalized four new sums of graphs and study the first Zagreb index and coindex of the resulting graphs. In addition, we give the short proof for the earlier results of Deng, Sarala, Ayyaswamy and Balachandran [Appl. Math. Comput. 275 (2016) 422-431] on the first Zagreb index of four operations on graphs by different approach.