The Szeged Index Of Some Nano Molecular Graph

A C₄C₈ net is a trivalent decoration made by alternating squares C4 and octagons C8.It can cover either a cylinder or a torus. The C₄C₈ net is called TUC₄C₈(R) nanotorus if itcan cover a torus. The C₄C₈ net is called TUC₄C₈(R) nanotorus if it can cover a cylinder.Some graph theory researcher and chemist have defined some topological index. In1947, The Wiener index as the first topological index was mentioned by chemist HaroldWiener. In 1994, the topological index W?(G) was mentioned by I. Gutman at the AttilaJozsef University in Szeged。Later put it named Szeged index and abbreviated as Sz(G).In the acyclic graph, the Value is equal for Sz(G) and W(G). But in the cyclic graph, therelationship of the Sz(G) and W(G) has not given conclusion. The edge Szeged index,Sze(G), is analogous to the Szeged index. In this paper, the Szeged index and edge Szegedindex of some C₄C₈ molecular are computed.The paper consists of five sections. In the first section, we introduced the research back-ground of nano molecular. Then we introduced some correlative definitions and notations oftopological index. In the second section, we characterized the relations between the Szegedindex and the vertex connection and obtain another formula of Szeged index. Besides, In thissection, we computed TUC₄C₈(S) nanotorus get the same conclusion by the definition ofSzeged index. In the third section, we compute the Szeged index of TUC₄C₈(R) nanotube.In the forth section, we introduced the edge Szeged index and given a fomula of the Sze(G)and computed the edge Szeged index of the TUC₄C₈(S) nanotube. In the last section, wegive a bound of the vertex PI index of connection graph.