8,666 research outputs found
The modified Schultz index of graph operations
Given a simple and connected graph G with vertex set V , denoting by dG(u) the degree of a vertex u and dG(u, v) the distance of two vertices, the modified Schultz index of G is given by S
∗
P
(G) =
{u,v}⊆V
dG(u) dG(v) dG(u, v), where the summation goes over all non ordered pairs of vertices of G. In
this paper we consider some graph operations, namely cartesian product, complete product, composition and
subdivision, and we obtain explicit formulae for the modified Schultz index of a graph in terms of the number
of vertices and edges as well as some other topological invariants such as the Wiener index, the Schultz index
and the first and second Zagreb indices
Moments in graphs
Let be a connected graph with vertex set and a {\em weight function}
that assigns a nonnegative number to each of its vertices. Then, the
{\em -moment} of at vertex is defined to be
M_G^{\rho}(u)=\sum_{v\in V} \rho(v)\dist (u,v) , where \dist(\cdot,\cdot)
stands for the distance function. Adding up all these numbers, we obtain the
{\em -moment of }: M_G^{\rho}=\sum_{u\in
V}M_G^{\rho}(u)=1/2\sum_{u,v\in V}\dist(u,v)[\rho(u)+\rho(v)]. This
parameter generalizes, or it is closely related to, some well-known graph
invariants, such as the {\em Wiener index} , when for every
, and the {\em degree distance} , obtained when
, the degree of vertex . 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
Computing Reformulated First Zagreb Index of Some Chemical Graphs as an Application of Generalized Hierarchical Product of Graphs
The generalized hierarchical product of graphs was introduced by L.
Barri\'ere et al in 2009. In this paper, reformulated first Zagreb index of
generalized hierarchical product of two connected graphs and hence as a special
case cluster product of graphs are obtained. Finally using the derived results
the reformulated first Zagreb index of some chemically important graphs such as
square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer
fullerene, zig-zag polyhex nanotube and dicentric dendrimers are computed.Comment: 12 page
The Eccentric Connectivity Polynomial of some Graph Operations
The eccentric connectivity index of a graph G, ξ^C, was proposed
by Sharma, Goswami and Madan. It is defined as ξ^C(G) =
∑ u ∈ V(G) degG(u)εG(u), where degG(u) denotes the degree of the vertex x
in G and εG(u) = Max{d(u, x) | x ∈ V (G)}. The eccentric connectivity
polynomial is a polynomial version of this topological index. In this paper,
exact formulas for the eccentric connectivity polynomial of Cartesian
product, symmetric difference, disjunction and join of graphs are presented.* The work of this author was supported in part by a grant from IPM (No. 89050111)
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