Sharp bounds for the second Zagreb index and co-index under trees of given conditions

Abstract

The second Zagreb index and co-index are two important graph invariants. They are defined as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is the complement of G and [Formula: see text] is the degree of a vertex u in G. In this paper, we study the upper and lower bounds for second Zagreb index and co-index of trees. Moreover, we determine the conditions that the upper and lower bounds for second Zagreb index and co-index of graphs are equal.

Keywords

• Second Zagreb index
• second Zagreb co-index
• trees
• diameter
• pendant vertices