The generalized Zagreb index is an extension of both ordinary and variable Zagreb indices. In this paper, we present exact formulae for the values of the generalized Zagreb index for product graphs. Results are applied to some graphs of general and chemical interest such as nanotubes and nanotori.

The first variable Zagreb index of graph G is defined as Ml1 (G)=SxÎv (G) d (v)2l, where l is a real number and d (v) is the degree of vertex v. In this paper, some lower and upper bounds for the expected value and distribution function of this index in random increasing trees (recursive trees, plane-oriented recursive trees and binary increasing trees) are given.

Let G (V,σ, ,μ, ) be a fuzzy graph of order n, where σ,(u) is the vertex membership, μ,(u,v) is membership value of an edge and μ,(u) is the strength of vertex. The _rst fuzzy Zagreb index is the sum σ,(ui) μ,(ui) + σ,(uj ) μ,(uj ) where uiuj ,μ,and the corresponding fuzzy Zagreb matrix is the square matrix of order n whose (I, j)th entry whenever i ≠,j, is σ,(ui) μ,(ui)+ σ,(uj ) μ,(uj ) and zero otherwise. In this paper, we introduce the Zagreb Estrada index of fuzzy graphs and establish some bounds for it.

‎Cruz‎, ‎Monsalve and Rada [Extremal values of vertex-degree-based topological indices of chemical trees‎, ‎Appl‎. ‎Math‎. ‎Comput‎. ‎380 (2020) 125281] posed an open problem to find the maximum value of the exponential second Zagreb index for chemical trees of given order‎. ‎In this paper‎, ‎we solve the open problem completely‎.

Let G be a simple connected graph. The first and second Zagreb indices have been introduced as M1(G)= åvÎV(G)degG(v)2 and M2(G)= åuvÎE(G)degG(u)degG(v), respectively, where degG v(degG u) is the degree of vertex v (u). In this paper, we define a new distance-based named Hyper-Zagreb as HM(G) = åe=uvÎE(G)(degG(u) + degG(v))2. In this paper, the Hyper-Zagreb index of the Cartesian product, composition, join and disjunction of graphs are computed.

‎Bucket recursive trees are an interesting and natural generalization of recursive trees‎. ‎In this model the nodes are buckets that can hold up to b≥ 1 labels‎. ‎The (modified) Zagreb index of a graph is defined as the sum of‎ ‎the squares of the outdegrees of all vertices in the graph‎. ‎We give the mean and variance of this index in random bucket recursive trees‎. ‎Also‎, ‎two limiting results on this index are given‎.

In this paper we study the Zagreb index in bucket recursive trees containing buckets with variable capacities. This model was introduced by Kazemi in 2012. We obtain the mean and variance of the Zagreb index and introduce a martingale based on this quantity.

In this paper, we used linear, quadratic, cubic, logarithmic, and exponential regression models to analyze the statistical properties of certain important molecular structures such as chloroquine, hydroxychloroquine, and remdesivir, taking into account various forms of augmented Zagreb indices. The research could be a fresh attempt to improve QSPR model prediction analysis using the above molecular descriptors, which are used to research chemical, medical, and pharmacological qualities.

For a graph G with edge set E (G), the multiplicative second Zagreb index of G is defined as Õ2(G)=ÕuvÎE(G) [dG (u) dG (v)], where dG (v) is the degree of vertex v in G. In this paper, we identify the eighth class of trees, with the rst through eighth smallest multiplicative second Zagreb indices among all trees of order n³14.

For a graph G, the third neighborhood degree index of G is defined as: ND3(G) = X ab∈E(G) δG(a)δG(b)  δG(a) + δG(b) , where δG(a) represents the sum of degrees of all neighboring vertices of vertex a. In this short paper, we establish a new lower bound on the third neighborhood degree index of trees and characterize the extremal trees achieving this bound.