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]

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

The double graph of a graph ${\displaystyle G}$, denoted ${\displaystyle {\mathcal {D}}[G]}$, has a known index in relation to ${\displaystyle G}$ itself:^[3]

${\displaystyle Sz({\mathcal {D}}[G])=16Sz(G)}$

References

1. Gutman, I. (1994), "A formula for the Wiener number of trees and its extension to graphs containing cycles", Graph Theory Notes, NY, 27: 9–15.
2. Khalifeh, M.H.; Darafsheh, M.R; Jolany, H. (2011), "The Wiener, Szeged, and PI Indices of a Dendrimer Nanostar", Journal of Computational and Theoretical Nanoscience, 8 (2): 220–223, Bibcode:2011JCTN....8..220K, doi:10.1166/jctn.2011.1681.
3. Ghasemi, Mehdi; Madanshekaf, Ali (27 September 2023). "On the Topological Indices on Double Graphs". Caspian Journal of Mathematical Sciences. 12 (2): 423–439. doi:10.22080/CJMS.2023.25624.1660.