On the zeroth-order general Randić index, variable sum exdeg index and trees having vertices with prescribed degree

Abstract

The zeroth-order general Randić index (usually denoted by 0R α) and variable sum exdeg index (denoted by SEia) of a graph G are defined as 0R α(G) =Σv V (G)(dv)α and SEia(G) =Σv V (G)dvadv, respectively, where dv is degree of the vertex v V (G), a is a positive real number different from 1 and α is a real number other than 0 and 1. A segment of a tree is a path P, whose terminal vertices are branching or/and pendent, and all non-terminal vertices (if exist) of P have degree 2. For n ≥ 6, let PTn,n1, n,k, n,b be the collections of all n-vertex trees having n1 pendent vertices, k segments, b branching vertices, respectively. in this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randić index and variable sum exdeg index are determined from the collections PTn,n1, STn,k, BTn,b. The obtained extremal trees for the collection STn,k are also extremal trees for the collection of all n-vertex trees having fixed number of vertices with degree 2 (because the number of segments of a tree T can be determined from the number of vertices of T having degree 2 and vice versa).