For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as M1(G) = ∑ v∈V (G) dG(v) 2 where dG(v) is the degree of vertex v in G. The alternative expression for M1(G) is ∑ uv∈E(G)(dG(u)+dG(v)). Very recently, Eliasi, Iranmanesh and Gutman [7] introduced a new graphical invariant ∏∗ 1(G) = ∏ uv∈E(G)(dG(u) + dG(v)) as the multiplicative version of ...

For a nontrivial graph G, its first Zagreb coindex is defined as the sum of degree sum over all non-adjacent vertex pairs in G and the second Zagreb coindex is defined as the sum of degree product over all non-adjacent vertex pairs in G. Till now, established results concerning Zagreb coindices are mainly related to composite graphs and extremal values of some special graphs. The existing liter...

zagreb indices belong to better known and better researched topological indices. weinvestigate here their ability to discriminate among benzenoid graphs and arrive at some quiteunexpected conclusions. along the way we establish tight (and sometimes sharp) lower andupper bounds on various classes of benzenoids.

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we investigate Zagreb indices of bicyclic graphs with a given matching number. Sharp upper bounds for the first and second Zagreb indices of bicyclic graphs in terms of the...

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we study the Zagreb indices of n-vertex connected graphs with k cut vertices, the upper bound for M1and M2-values of n-vertex connected graphs with k cut vertices are deter...

For a graph $G$ with edge set $E(G)$, the multiplicative sum Zagreb index of $G$ is defined as$Pi^*(G)=Pi_{uvin E(G)}[d_G(u)+d_G(v)]$, where $d_G(v)$ is the degree of vertex $v$ in $G$.In this paper, we first introduce some graph transformations that decreasethis index. In application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum Zagreb ...

‎for a graph $g$ with edge set $e(g)$‎, ‎the multiplicative second zagreb index of $g$ is defined as‎ ‎$pi_2(g)=pi_{uvin e(g)}[d_g(u)d_g(v)]$‎, ‎where $d_g(v)$ is the degree of vertex $v$ in $g$‎. ‎in this paper‎, ‎we identify the eighth class of trees‎, ‎with the first through eighth smallest multiplicative second zagreb indeces among all trees of order $ngeq 14$‎.

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments of Zn, the Zagreb index of a random recursive tree of size n, are obtained. We also show that the random process {Zn − E[Zn], n ≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by...

todeschini et al. have recently suggested to consider multiplicative variants of additive graphinvariants, which applied to the zagreb indices would lead to the multiplicative zagrebindices of a graph g, denoted by ( ) 1  g and ( ) 2  g , under the name first and secondmultiplicative zagreb index, respectively. these are define as  ( )21 ( ) ( )v v gg g d vand ( ) ( ) ( )( )2 g d v d v gu...

The Zagreb indices are among the oldest and the most famous topological molecular structure-descriptors. The first Zagreb index is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper, we characterize the extremal graphs with maximal, sec...