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 Cet article a éte moissonné depuis la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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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 
@article{BASS_2005_30_1_a0,
     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},
     year = {2005},
     volume = {30},
     number = {1},
     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/