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Theoretical prediction of chemical properties via mathematics and discovery of extremal configurations

The motivation The Atom-Bond Connectivity index (ABC-index) is a mathematical descriptor that characterizes molecular bonds based on unique aspects of the topological structure. This approach uses graph theory, which is a branch of mathematics, to enable mathematical modelling of molecules. The ABC-index has diverse chemical applications, especially in chemical thermodynamics; for example, it is related to heat formation and stabilization in alkanes. The value of the ABC-index can be computed by considering the molecular graph, which can be thought of as a topological or three-dimensional layout of chemical bonds between the atoms forming a molecule. Based on the resulting value, one can predict – without actually producing the molecule – what kind of properties the compound will have. In using the ABC-index to design new substances with useful properties, a key step is to find out the kind of chemical bonds that give an extremal value (i.e., the largest or smallest value possible). This approach to predicting the properties of a compound raises purely mathematical questions, some of which are easy to answer and others that are notoriously difficult to solve.

Recent work by other researchers, including Lin et al., gave rise to a proposed mathematical statement (a conjecture) about the minimum possible ABC-index of trees (acyclic graphs) with a fixed number of leaves (nodes with only one bond). Their predictions, which were based on mathematical analysis and extensive computer calculations, gave answers for trees having up to 100 leaves. 

The discovery – In examining this problem, Dr. Bojan Mohar (Dept. of Mathematics, Simon Fraser University) discovered that new unexpected phenomena arise when the number of nodes gets larger.  His recent paper presents how he reached the correct answer to this mathematical challenge and in so doing, the conjecture of Lin et al. has been refuted.  In Dr. Mohar’s work, it is shown that for trees with t leaves, the extremal tree T(t) is unique if t > 1194.  The number of nodes of T(t) is t + [t/10] +1 (when t modulo 10 is between 0 and 4 or when it is 5, 6, or 7 and t is sufficiently large).  The number of nodes is one higher when t modulo 10 is equal to 8 or 9.  The structure of these trees gives precise asymptotics for their ABC-index.

Its significance – While the obtained results are purely theoretical, they give deeper insight into properties of the ABC-index, and the extremal examples give reference values for comparison and analysis of this quantity when studied on large molecular graphs.

Read the paper“The Structure of ABC-Minimal Trees with Given Number of Leaves” by Mohar, B. MATCH – Communications in Mathematical and in Computer Chemistry 79(2):415-431 (2018)

Website article compiled by Jacqueline Watson with Theresa Kitos