The minimal-ABC trees with $$B_2$$-branches

Zhibin Du,Darko Dimitrov

doi:10.1007/s40314-020-1119-7

Abstract

The atom–bond connectivity (ABC) index is a degree-based graph topological index with a lot of chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. It is known (Chen and Guo in MATCH Commun Math Comput Chem 65:713–722, 2011; Das et al. in MATCH Commun Math Comput Chem 76:159–170, 2011) that among all connected graphs, trees minimize the ABC index (such trees are called minimal-ABC trees). Several structural properties of minimal-ABC trees were proved in the past several years. Here we continue to make a step forward towards the complete characterization of the minimal-ABC trees. In Dimitrov (Discrete Appl Math 204:90–116, 2016), it was shown that a minimal-ABC tree cannot contain more than 11 so-called $$B_2$$-branches. We improve this result by showing that if a minimal-ABC tree of order larger than 39 contains so-called $$B_1$$-branches, then it contains exactly one $$B_2$$-branch, and if a minimal-ABC tree of order larger than 163 contains no $$B_1$$-branch, then it contains at most two $$B_2$$-branches.

Highlights

The atom-bond connectivity index, widely known as ABC index, of a graph is a thoroughly studied vertex-degree-based graph invariant both in chemistry and mathematical communities
P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G = (V, E), the ABC index of G is defined as ABCðGÞ 1⁄4 ij2E ðdi þ dj À 2Þ=ðdidjÞ, where di denotes the degree of the vertex i, and ij is the edge incident to the vertices i and j
The applicability of the ABC index in chemical thermodynamics and other areas of chemistry, such as in dendrimer nanostars, benzenoid systems, fluoranthene congeners, and phenylenes is well studied in the literature

Summary

Introduction

The atom-bond connectivity index, widely known as ABC index, of a graph is a thoroughly studied vertex-degree-based graph invariant both in chemistry and mathematical communities. The relevance of the ABC index, in what we call today chemical graph theory, was first revealed two decades ago by Estrada, Torres, Rodrıguez, and Gutman in [2]. Estrada [3] uncovered the significance of ABC index on the stability of branched alkanes, based on at that time a novel quantum-theory-like exposition. These studies were the trigger point for an uncountable number of papers on a new found area: chemical graph theory. Another example of the importance of this topological descriptor can be seen on the calculation of the ABC index of an infinite class of nanostar dendrimers, artificially manufactured or synthesized molecule built up from branched units called monomers [6]

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Conclusion