Geodesic
Some relations between distance-based polynomials of trees
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 30 (2005) no. 1.

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

The Hosoya polynomial $H(G,\lambda)$ of a graph $G$ has the property that its first derivative at $\lambda=1$ is equal to the Wiener index. Sometime ago two distance-based graph invariants were studied -- the Schultz index $S$ and its modification $S^\ast$ . We construct distance--based graph polynomials $H_1(G,\lambda)$ and $H_2(G,\lambda)$ , such that their first derivatives at $\lambda=1$ are, respectively, equal to $S$ and $S^\ast$ . In case of trees, $H_1(G,\lambda)$ and $H_2(G,\lambda)$ are related with $H(G,\lambda)$ .
Keywords: Graph polynomial, distance (in graph), tree, Wiener index, Schultz index 
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     author = {I. Gutman},
     title = {Some relations between distance-based polynomials of trees},
     journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
     pages = {1 - 7},
     publisher = {mathdoc},
     volume = {30},
     number = {1},
     year = {2005},
     zbl = {1120.05304},
     url = {http://geodesic.mathdoc.fr/item/BASS_2005_30_1_a0/}
}
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I. Gutman. Some relations between distance-based polynomials of trees. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 30 (2005) no. 1. http://geodesic.mathdoc.fr/item/BASS_2005_30_1_a0/