The Nordhaus–Gaddum-type inequalities for the Zagreb index and co-index of graphs

Let k ≥ 2 be an integer, a k -decomposition ( G 1 , G 2 , … , G k ) of the complete graph K n is a partition of its edge set to form k spanning subgraphs G 1 , G 2 , … , G k . In this contribution, we investigate the Nordhaus–Gaddum-type inequality of a k -decomposition of K n for the general Zagreb index and a 2 -decomposition for the Zagreb co-indices, respectively. The corresponding extremal graphs are characterized.