On Segments, Vertices of Degree Two and the First Zagreb Index of Trees

The rst Zagreb index M1 of a graph G is equal to the sum of squares of the vertex degrees of G. A segment of a tree is a path-subtree whose terminal vertices are branching or pendent vertices. In this paper, we characterize the trees which minimize and maximize the rst Zagreb index among all trees with xed number of segments, respectively. As a byproduct, we also prove that these trees also share the minimum and maximum rst Zagreb index among all trees with xed number of vertices of degree two, respectively.