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 R 0 α ) and variable sum exdeg index (denoted by SEI a ) of a graph G are defined as R 0, 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 pendent, and all non-terminal vertices (if exist) of P have degree 2. For n ≥ 6, let PT n,n1 , ST n,k , BT n,b be the collections of all n-vertex trees having n 1 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 PT n,n1 , ST n,k , BT n,b . The obtained extremal trees for the collection ST n,k are also extremal trees for the collection of all n-vertex trees having fixed number of vertices with degree 2 (because it is already known that the number of segments of a tree T can be determined from the number of vertices of T with degree 2 and vise versa).

On the extremal cactus graphs for variable sum exdeg index with a fixed number of cycles

“…Ali and Dimitrov [2] gave alternative proofs of four main results proved in [7] and extended one of the theorems proved in [7]. Khalid and Ali [10] investigated the properties of the variable sum exdeg index SEI a of trees having a fixed number of vertices and vertices with a prescribed degree. Recently, Dimitrov and Ali [5] attempted the problem of finding graphs with a fixed number of vertices and cyclomatic number (the minimum number of edges of a graph whose removal give a graph containing no cycle).…”

Zeroth-order general Randi´c index of trees

“…Khalid and Ali [7] studied tress of given order and given number of leaves/segments/branching vertices and they determined the maximum and minimum zeroth-order general Randić index for those trees. Yamaguchi [12] obtained the largest zeroth-order general Randić index for trees of given order, and given diameter or radius.…”

On the variable sum exdeg index/coindex of graphs

“…For a > 1, the results of [1,8] were generalized by Dimitrov and Ali [5] by considering the graphs of a fixed order and cyclomatic number; in [5], the case 0 < a < 1 was also discussed but only the partial solutions to the considered problems for this certain case were obtained. The problem of finding graphs having the extremum values of the variable sum exdeg index of the trees of a fixed order and with the vertices having prescribed degrees was attacked in [11]. Additional recent results about the variable sum exdeg index can be found in the papers [3,7,9,13,18].…”