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

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