Remarks on Multiplicative Atom-Bond Connectivity Index

The atom-bond connectivity (ABC) index is one of the most actively studied degree-based graph invariants that are found in a vast variety of chemical applications. This paper is devoted to establishing some extremal results regarding the variant of the ABC index, the so-called multiplicative ABC index (ABC \Pi ), which, for a graph G , is defined as {{ ABC}}\Pi (G)=\prod _{uv\in E(G)}\sqrt {\frac {d(u)+d(v)-2}{d(u)d(v)}} . We have shown that the complete graph K_{n} has a minimum ABC \Pi index among connected simple graphs with n vertices, while the star graph S_{n-1} has the maximum ABC \Pi index. As S_{n-1} attains the maximum amongst bipartite graphs on n vertices, we additionally show that the bipartite complete balanced graph K_{\left \lfloor{ n/2 }\right \rfloor, \left \lceil{ n/2 }\right \rceil } attains the minimum in this class of graphs. As an interesting problem, we propose to characterize the trees with the minimum value of this index, and, here, we have some structural properties of these trees. We conclude this paper with few conjectures for possible further work.