On the Extremal Zagreb Indices of Trees with Given Number of Segments or Given Number of Branching Vertices

The first Zagreb index M-1 of a graph is equal to the sum of squares of its vertex degrees, and the second Zagreb index M-2 is equal to the sum of products of degrees of pairs of adjacent vertices. A vertex of a tree with degree at least three is called a branching vertex and a segment of a tree is a path-subtree whose terminal vertices are branching or pendent vertices. Sharp lower and upper bounds on the second Zagreb index of trees with fixed number of segments are determined and the corresponding extremal trees are characterized. As the number of segments in a tree is determined by the number of vertices of degree two (and vice versa) in this way the extremal trees with prescribed number of vertices of degree two whose second Zagreb index is minimum (or maximum) are determined, too. Also, sharp lower and upper bounds on Zagreb indices M-1 and M-2 of n-vertex trees with given number of branching vertices are determined, and corresponding extremal trees are characterized