Szeged index

In chemical graph theory, the Szeged index is a topological index of a molecule, used in biochemistry. The Szeged index, introduced by Iván Gutman, ^[1] generalizes the concept of the Wiener index introduced by Harry Wiener. The Szeged index of a connected graph G is defined as

${\displaystyle Sz(G)=\sum _{e\in E(G)}n_{1}(e\mid G)n_{2}(e\mid G),}$

If e is an edge of G connecting vertices u and v, then we write e = uv or e = vu. For ${\displaystyle e=uv\in E(G)}$, let ${\displaystyle n_{1}(e\mid G)}$ and ${\displaystyle n_{2}(e\mid G)}$ be respectively the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u.

Szeged index plays an important role in information theory. One way to measure a network structure is through the so-called topological indices. Szeged index has been shown to correlate well with numerous biological and physicochemical properties.

Examples[]

The Szeged index of Dendrimer Nanostar of the following figure can be calculated by^[2]

${\displaystyle Sz(T_{n})=1620n\cdot 4^{n}-2376\cdot 4^{n}+2862\cdot 2^{n}-432,\quad n\geq 0.}$

References[]