Extremal Mostar indices of tree-like polyphenyls

摘要

Given a graph G, the Mostar index Mo(G) is the sum of absolute values of the differences between n(u)(e) and n(v)(e) over all edges e = uv of G, where n(u)(e) and n(v)(e) are, respectively, the number of vertices of G lying closer to u than to v and the number of vertices of G lying closer to v than to u. A tree-like polyphenyl is a polycyclic aromatic hydrocarbon consisting of benzene rings, whose chemical graph will tend to a tree after contracting each hexagon into a vertex (the tree is called a contracted tree). A polyphenyl chain is a tree-like polyphenyl whose contracted tree is a path. In this paper, those polyphenyl chains with n benzene rings having the least, the second least, the greatest and the second greatest Mostar indices are determined, respectively. Those tree-like polyphenyls with n benzene rings having the least and the second least Mostar indices are also identified. What is more, some properties of those tree-like polyphenyls with n benzene rings having the greatest Mostar index are obtained. At the end we state some further research problems.