On the maximum augmented Sombor index of molecular trees with a perfect matching

Further results on atom-bond connectivity index of trees

The reduced Sombor index and the exponential reduced Sombor index of a molecular tree

The Randic index R(G) of a graph G is the sum of the weights (dudv)-1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randic indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex trees with the third for n ? 7, the fourth for n ? 10, the fifth and the sixth for n ? 11 maximum Randic indices. Recently, Li et al. [The Randic indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex trees with the seventh, the eighth, the ninth and the tenth for n ? 11 maximum Randic indices. In this paper, we correct the ordering for the Randic indices of trees obtained by Li et al., and characterize the trees with from the seventh to the sixteenth maximum Randic indices. The obtained extremal trees are molecular and thereby the obtained ordering also holds for molecular trees.

Maximum Atom Bond Sum Connectivity Index of Molecular Trees with a Perfect Matching

Let G be a simple graph with n vertices and let λ1, λ2, . . . , λ n be the eigenvalues of its adjacency matrix. The Estrada index of G is a recently introduced molecular structure descriptor, defined as $${EE (G) = \sum_{i = 1}^n e^{\lambda_i}}$$ , proposed as a measure of branching in alkanes. In order to support this proposal, we prove that among the trees with fixed maximum degree Δ, the broom B n,Δ, consisting of a star S Δ+1 and a path of length n−Δ−1 attached to an arbitrary pendent vertex of the star, is the unique tree which minimizes even spectral moments and the Estrada index, and then show the relation EE(S n ) = EE(B n,n−1) > EE(B n,n−2) > . . . > EE(B n,3) > EE(B n,2) = EE(P n ). We also determine the trees with minimum Estrada index among the trees with perfect matching and maximum degree Δ. On the other hand, we strengthen a conjecture of Gutman et al. [Z. Naturforsch. 62a (2007), 495] that the Volkmann trees have maximal Estrada index among the trees with fixed maximum degree Δ, by conjecturing that the Volkmann trees also have maximal even spectral moments of any order. As a first step in this direction, we characterize the starlike trees which maximize even spectral moments and the Estrada index.

Complete solution to open problems on exponential augmented Zagreb index of chemical trees

Open problems on the exponential vertex-degree-based topological indices of graphs

Extremal augmented Zagreb index of trees with given numbers of vertices and leaves

Augmented Zagreb index of trees and unicyclic graphs with perfect matchings

The first general multiplicative Zagreb index of a graph G is defined as $$P_1^a (G) = \prod _{v \in V(G)} (deg_G (v))^a$$ and the second general multiplicative Zagreb index is $$P_2^a (G) = \prod _{v \in V(G)} (deg_G (v))^{a \, deg_G (v)}$$ , where V(G) is the vertex set of G, $$deg_{G} (v)$$ is the degree of v in G and $$a \ne 0$$ is a real number. We present lower and upper bounds on the general multiplicative Zagreb indices for trees and unicyclic graphs of given order with a perfect matching. We also obtain lower and upper bounds for trees and unicyclic graphs of given order and matching number. All the trees and unicyclic graphs which achieve the bounds are presented, thus our bounds are sharp. Bounds for the classical multiplicative Zagreb indices are special cases of our theorems and those bounds are new results as well.

The Symmetric division deg (SDD) index is a well-established valuable index in the analysis of quantitative structure-property and structure-activity relationships for molecular graphs. In this paper, we study the range of SDD-index for special classes of trees and unicyclic graphs. We present the first four lower bounds for SDD-index of trees and unicyclic graphs, which admit a perfect matching and find the subclasses of graphs that attain these bounds. Further, we also compute the upper bounds of SDD-index for the collection of molecular graphs, namely the trees and unicyclic graphs, each having maximum degree four and that admit a perfect matching.

The Randic index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. We give a sharp lower bound on the Randic index of conjugated trees (trees with a perfect matching) in terms of the number of vertices. A sharp lower bound on the Randic index of trees with a given size of matching is also given

For a graph G, the vertex-degree function index of G is defined as Hf (G) = ?u?V(G) f(degG(u)), where degG(u) stands for the degree of vertex u in G and f(x) is a function defined on positive real numbers. In this article, we determine the extremal values of the vertex-degree function index of trees with given number of pendent vertices/segments/branching vertices/maximum degree vertices and with a perfect matching when f(x) is strictly convex (resp. concave). Moreover, we use the results directly to some famous topological indices which belong to the type of vertex-degree function index, such as the zerothorder general Randic index, sum lordeg index, variable sum exdeg index, Lanzhou index, first and second multiplicative Zagreb indices.

Maximal augmented Zagreb index of trees with given diameter

The harmonic index of a graph G is defined as the sum of weights 2/d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u in G. In this paper, some general properties of the harmonic index for molecular trees are explored. Moreover, the smallest and largest values of harmonic index for molecular trees with given pendent vertices are provided, respectively.