Extremal bi-Wiener indices of unbranched catacondensed benzenoids and polyphenyl hexagonal chains

The bi-Wiener index is a recently introduced distance-based invariant for connected bipartite graphs. Let $G$ be a connected bipartite graph with bipartition $V(G)=X\cup Y$. The bi-Wiener index of $G$ is defined by

$$W_b(G)=\sum _{(u,v)\in X\times Y}{d}_G(u,v),$$

where $d_G(u,v)$ is the distance between $u$ and $v$ in $G$. In this paper, we study extremal bi-Wiener indices of two families of hexagon-based molecular chains. The first family considered here is the family of unbranched catacondensed benzenoids, also called hexagonal chains, in which consecutive hexagons share an edge. If $F_n\in {\mathcal{U}}_n$ is an unbranched catacondensed benzenoid with $n$ hexagons, then

$$\frac{4n^3+36n^2-10n+15}{3}\le W_b(F_n)\le \frac{8n^3+18n^2+16n+3}{3}.$$

For $n\ge 3$, the lower bound is attained uniquely by the helicene chain, and the upper bound is attained uniquely by the linear chain. The second family is the family of polyphenyl hexagonal chains, in which consecutive hexagons are joined by a cut edge. If $G_n\in {\mathcal{P}}_n$ is a polyphenyl hexagonal chain with internal configuration sequence ${\ell }_2,{\ell }_3,\dots ,{\ell }_{n-1}\in \{1,2,3\}$, then

$$W_b(G_n)=3n^3+27n^2-15n+18\sum _{k=2}^{n-1}{\ell }_k(k-1)(n-k),$$

where the sum is empty if $n\le 2$. Consequently, among all polyphenyl hexagonal chains with $n$ hexagons, the ortho-chain has the minimum bi-Wiener index and the para-chain has the maximum bi-Wiener index.