Let G be a connected graph with vertex set V and a weight function that assigns
a nonnegative number to each of its vertices. Then, the -moment of G at vertex u
is de ned to be M
G(u) =
P
v2V (v) dist(u; v), where dist( ; ) stands for the distance
function. Adding up all these numbers, we obtain the -moment of G:
This parameter generalizes, or it is closely related to, some well-known graph invari-
ants, such as the Wiener index W(G), when (u) = 1=2 for every u 2 V , and the
degree dista...
Let G be a connected graph with vertex set V and a weight function that assigns
a nonnegative number to each of its vertices. Then, the -moment of G at vertex u
is de ned to be M
G(u) =
P
v2V (v) dist(u; v), where dist( ; ) stands for the distance
function. Adding up all these numbers, we obtain the -moment of G:
This parameter generalizes, or it is closely related to, some well-known graph invari-
ants, such as the Wiener index W(G), when (u) = 1=2 for every u 2 V , and the
degree distance D0(G), obtained when (u) = (u), the degree of vertex u.
In this paper we derive some exact formulas for computing the -moment of a
graph obtained by a general operation called graft product, which can be seen as a
generalization of the hierarchical product, in terms of the corresponding -moments
of its factors. As a consequence, we provide a method for obtaining nonisomorphic
graphs with the same -moment for every (and hence with equal mean distance,
Wiener index, degree distance, etc.). In the case when the factors are trees and/or
cycles, techniques from linear algebra allow us to give formulas for the degree distance
of their product.
This parameter generalizes, or it is closely related to, some well-known graph invariants, such as the Wiener index W (G), when rho(u) = 1/2 for every u is an element of V, and the degree distance D'(G), obtained when rho(u) = delta(u), the degree of vertex u. In this paper we derive some exact formulas for computing the rho-moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding rho-moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same rho-moment for every rho (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product.
Citation
Dalfo, C.; Fiol, M.; Garriga, E. Moments in graphs. "Discrete applied mathematics", 2013, vol. 161, núm. 6, p. 768-777.