Complexity of the Szeged index, edge orbits, andsome nanotubical fullerenes

Abstract

LetIbe a summation-type topological index. TheI-complexityCI(G)of a graphGisthe number of different contributions toI(G)in its summation formula. In this paper thecomplexity$C_{Sz}$(G)is investigated, whereSzis the well-studied Szeged index. Let$O_e$(G)(resp.$O_v$(G)) be the number of edge (resp. vertex) orbits ofG. While$C_{Sz}$(G)≤$O_e$(G)holds for any graphG, it is shown that for anym≥1there exists a vertex-transitive graphGmwithCSz(Gm) =Oe(Gm) =m. Also, for any1≤k≤m+ 1there exists a graphGm,kwith$C_{Sz}$($Gm,_k$) =$O_e$($Gm_k$) =mandCW($Gm,_k$) =Ov($Gm,_k$) =k. The Sz-complexity isdetermined for a family of (5,0)-nanotubical fullerenes and the Szeged index is comparedwith the total eccentricity.