The Szeged index, also known as the edge-Szeged index, is a graph index defined for a connected graph by
where
is the number of vertices closer to than to , and is defined analogously (Gutman 1994; Klavžar et
al. 1996; Devillers and Balaban 1999, p. 257).
The Szeged index generalizes the Wiener index to graphs with cycles. For a tree, it is equal to the Wiener
index.
The Szeged index of a graph or molecule is implemented in the Wolfram Function Repository as ResourceFunction["SzegedIndex"][g]. For molecules,
hydrogen atoms are ignored by default (Mangaldan).
Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands:
Gordon and Breach, pp. 42, 256-257, and 780, 1999.Gutman, I. "A
Formula for the Wiener Number of Trees and Its Extension to Graphs Containing Cycles."
Graph Theory Notes of New York27, 9-15, 1994.Gutman,
I. and Dobrynin, A. A. "The Szeged Index--A Success Story." Graph
Theory Notes of New York34, 37-44, 1998.Klavžar, S.;
Rajapakse, A.; and Gutman, I. "The Szeged and the Wiener Index of Graphs."
Appl. Math. Lett.9, 45-49, 1996. https://doi.org/10.1016/0893-9659(96)00071-7. Mangaldan, J. "SzegedIndex." Wolfram Function Repository. https://resources.wolframcloud.com/FunctionRepository/resources/SzegedIndex/.
by
is the number of vertices closer to 
than to 
, and 
is defined analogously (Gutman 1994; Klavžar et
al. 1996; Devillers and Balaban 1999, p. 257).
Mangaldan, J. "SzegedIndex." Wolfram Function Repository.