Abstract
The atom-bond connectivity index (or, for short, ABC index) is a molecular structure descriptor bridging chemistry to graph theory. It is probably the most studied topological index among all numerical parameters of a graph that characterize its topology. For a given graph G = (V, E), the ABC index of G is defined as , where di denotes the degree of the vertex i, and ij is the edge incident to the vertices i and j. A combination of physicochemical and the ABC index properties are commonly used to foresee the bioactivity of different chemical composites. Additionally, 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. While finding of the graphs with the greatest ABC-value is a straightforward assignment, the characterization of the tree(s) with minimal ABC index is a problem largely open and has recently given rise to numerous studies and conjectures. A B1-branch of a graph is a pendent path of order 2. In this paper, we provide an important step forward to the full characterization of these minimal trees. Namely, we show that a minimal-ABC tree contains neither 4 nor 3 B1-branches. The case when the number of B1-branches is 2 is also considered.
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
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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