Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes

Let $I$ be a summation-type topological index. The $I$-complexity $C_I(G)$ of a graph $G$ is the number of different contributions to $I(G)$ in its summation formula. In this paper the complexity $C_{Sz}(G)$ is investigated, where Sz is the well-studied Szeged index. Let $O_e(G)$ (resp. $O_v(G)$) be the number of edge (resp. vertex) orbits of $G$. While $C_{Sz}(G) \leq O_e(G)$ holds for any graph $G$, it is shown that for any $m\geq 1$ there exists a vertex-transitive graph $G_m$ with $C_{Sz}(G_m) = O_e(G_m) = m$. Also, for any $1\leq k\leq m+1$ there exists a graph $G_{m,k}$ with $C_{Sz}(G_{m,k}) = O_e(G_{m,k}) = m$ and $C_{W}(G_{m,k}) = O_v(G_{m,k}) = k$. The Sz-complexity is determined for a family of (5,0)-nanotubical fullerenes and the Szeged index is compared with the total eccentricity.