On two degree-and-distance-based graph invariants
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 41 (2016), p. 21
Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts
Let $G$ be a connected graph with vertex set
$V(G)$. For $u,v \in V(G)$, by $d(v)$ and $d(u,v)$ are denoted the
degree of the vertex $v$ and the distance between the vertices $u$
and $v$. A much studied degree--and--distance--based graph
invariant is the degree distance, defined as
$DD=\sum_{\{u,v\}\subseteq V(G)} [d(u)+d(v)]\,d(u,v)$. A related
such invariant is $ZZ=\sum_{\{u,v\}\subseteq V(G)} [d(u) \times
d(v)]\,d(u,v)$. If $G$ is a tree, then both $DD$ and $ZZ$ are
linearly related with the Wiener index $W = \sum_{\{u,v\}\subseteq
V(G)} d(u,v)$. We show how these relations can be extended in the
case when $d(u)$ and $d(v)$ are replaced by $f(u)$ and $f(v)$,
where $f$ is any function of the corresponding vertex. We also
give a few remarks concerning the discovery of $DD$ and $ZZ$.
Classification :
05C07, 05C12, 05C90
Keywords: degree (of vertex), distance (in graph), degree distance (of graph), Gutman index
Keywords: degree (of vertex), distance (in graph), degree distance (of graph), Gutman index
Ivan Gutman. On two degree-and-distance-based graph invariants. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 41 (2016), p. 21 . http://geodesic.mathdoc.fr/item/BASS_2016_41_a1/
@article{BASS_2016_41_a1,
author = {Ivan Gutman},
title = {On two degree-and-distance-based graph invariants},
journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
pages = {21 },
year = {2016},
volume = {41},
language = {en},
url = {http://geodesic.mathdoc.fr/item/BASS_2016_41_a1/}
}