On the difference between atom‐bond connectivity index and Randić index of binary and chemical trees

Abstract

This article is devoted to establishing some extremal results with respect to the difference of two well‐known bond incident degree indices [atom‐bond connectivity (ABC) index and Randić (R) index] for the chemical graphs representing alkanes. More precisely, the first three extremal trees with respect to ABC – R are characterized among all n‐vertex binary trees (the trees with maximum degree at most 3). The n‐vertex chemical trees (the trees with maximum degree at most 4) having the first three maximum ABC – R values are also determined.

Some new upper bounds for the inverse sum indeg index of graphs

“…In [31], finding best possible upper bound on the aforementioned invariant for molecular trees (graphs representing alkanes) was left as an open problem. In this section, we will see that this problem can be easily solved for the case of binary trees [1] (that is, the trees with maximum degree at most 3). Binary trees actually form a subclass of the class of all molecular trees.…”

“…We solve the system of Equations (21)- (25) for the unknowns n 1 , n 2 , n 3 , x 1,3 , x 3,3 . The values of x 1,3 and x 3,3 are given [1] below:…”

Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index

“…Nowadays, studying the relationship or comparison between topological indices, see [20][21][22][23], is becoming popular. Recently, Ali and Du [24] investigated extremal binary and chemical trees results for the difference between ABC and R indices. A tree with maximum degree at most three or four called a binary and chemical tree, respectively.…”

“…For a connected graph G of order at least 3, the difference between ABC and R is represented as (see [24])…”

“…In this paper, motivated by the results in [24], we further investigated the extremal chemical trees for ABC − R. Moreover, maximal trees with fixed number of pendant vertices are also investigated for ABC − R index. The techniques used in this paper are very similar to that of Refs.…”