On the Minimum ABC Index of Chemical Trees

The Atom Bond Connectivity index, also known as ABC index was defined by Estrada(4) with relation to the energy of formation of alkanes. It was quickly recognized that this index reflects important structural properties of graphs in general. The ABC index was extensively studied in the last three years, from the point of view of chemical graph theory(5,6), and in general graphs(1). It was also compared to other structural indices of graphs(2). Das derives multiple results with implications to the minimum/maximum ABC index on graphs. With relation to trees, it is known that among all the trees of the same number of vertices, the maximum ABC index is attained for the star graph. However, it is not known which tree(s) minimize(s) the ABC index. The problem seems to be hard. It is partially addressed in many sources(5,1,6), but remains open. In this paper we further investigate the trees that minimize the ABC index. Our investigations are limited to chemical trees, i.e. trees in which the maximum vertex degree is 4. The chemical trees were introduced to reflect the structure of the carbon chains and the molecules based on them. Our approach is algorithmic. We identify certain types of edges (chemical bonds) that are important and occur frequently in chemical trees. Further, we study how the removal of a certain edge, the intro- duction of certain edge or the contraction of certain edge affects the ABC-index of the tree. We pay particular attention to the examples of minimal ABC index chemical trees provided by Dimitrov(3). This paper will discuss the Atom-bond connectivity index, its origins, its uses and its applications. The central topics of discussion for this paper will include: an investigation of the Furtula examples (provided through a personal communica- tion), an exploration of the structure of the maximal and minimal chemical trees for the ABC index, and the contri- bution of certain types of vertices and edges to the ABC index of the graph as a whole. It will also consider graphs from a specific class Γ (to be defined later) and their ABC indices.