‎The first variable Zagreb index of graph $G$ is defined as‎ ‎begin{eqnarray*}‎ ‎M_{1,lambda}(G)=sum_{vin V(G)}d(v)^{2lambda}‎, ‎end{eqnarray*}‎ ‎where $lambda$ is a real number and $d(v)$ is the degree of‎ ‎vertex $v$‎. ‎In this paper‎, ‎some upper and lower bounds for the distribution function and expected value of this index in random increasing trees (rec...

Todeschini et al. have recently suggested to consider multiplicative variants of additive graph invariants, which applied to the Zagreb indices would lead to the multiplicative Zagreb indices of a graph G, denoted by ( ) 1  G and ( ) 2  G , under the name first and second multiplicative Zagreb index, respectively. These are define as     ( ) 2 1 ( ) ( ) v V G G G d v and ( ) ( ) ( ) ( ) 2...

The first Zagreb index of a graph G, with vertex set V (G) and edge set E(G), is defined as M1(G) = ∑ u∈V (G) d(u) 2 where d(u) denotes the degree of the vertex v. An alternative expression for M1(G) is ∑ uv∈E(G)[d(u) + d(v)]. We consider a multiplicative version of M1 defined as Π∗1(G) = ∏ uv∈E(G)[d(u) + d(v)]. We prove that among all connected graphs with a given number of vertices, the path ...

The variable Zagreb (v)M(2) index is introduced and applied to the structure-boiling point modeling of benzenoid hydrocarbons. The linear model obtained (the standard error of estimate for the fit model S(fit)=6.8 degrees C) is much better than the corresponding model based on the original Zagreb M2 index (S(fit)=16.4 degrees C). Surprisingly,the model based on the variable vertex-connectivity ...

There is a natural linkage between the molecular structures and the bio-medical and pharmacology characteristics. A topological index can be considered as transformation of chemical structure in to real number and has been used as a predictor parameter. There are certain vertex-degree-based topological indices which has been used extensively in the chemical graph theory but recently no further ...

For a simple graph G with n vertices and m edges, the first Zagreb index and the second Zagreb index are defined as M1(G) = ∑ v∈V d(v) 2 and M2(G) = ∑ uv∈E d(u)d(v). In [34], it was shown that if a connected graph G has maximal degree 4, then G satisfies M1(G)/n = M2(G)/m (also known as the Zagreb indices equality) if and only if G is regular or biregular of class 1 (a biregular graph whose no ...

Let [Formula: see text] be a graph. A set [Formula: see text] is a distance k-dominating set of G if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text], where k is a positive integer. The distance k-domination number [Formula: see text] of G is the minimum cardinality among all distance k-dominating sets of G. The first Zagreb index of G is defined as ...

In this paper, we present sharp bounds for the Zagreb indices, Harary index and hyperWiener index of graphs with a given matching number, and we also completely determine the extremal graphs. © 2010 Elsevier Ltd. All rights reserved.

By d(v|G) and d_2(v|G) are denoted the number of first and second neighborsof the vertex v of the graph G. The first, second, and third leap Zagreb indicesof G are defined asLM_1(G) = sum_{v in V(G)} d_2(v|G)^2, LM_2(G) = sum_{uv in E(G)} d_2(u|G) d_2(v|G),and LM_3(G) = sum_{v in V(G)} d(v|G) d_2(v|G), respectively. In this paper, we generalizethe results of Naji et al. [Commun. Combin. Optim. ...