The 2008 Wenchuan earthquake destroyed large areas of vegetation in the Baisha River and Longxi River basins, in Dujiangyan County, China. There were several debris flow events in these mountainous river basins after 2008. Currently, these damaged vegetation areas are in various stages of recovery. This recovery vegetation improves the resistance of slopes to both surficial erosion and mass was...

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal ABC index is still an open problem. In [14], it was shown that trees with minimal ABC index do not contain so-called Bk-branches, with k ≥ 5, and that they do not have more than four B4-branches. Our main results here r...

We present a theory of topological edge states in onedimensional resonant photonic crystals with compound unit cell. We demonstrate how the structure, despite being onedimensional, can be characterized by topological indices. Contrary to conventional electronic topological states the modes under consideration are radiative, i.e., they decay in time due to the light escape through the structure ...

Mobeen Munir 1, Waqas Nazeer 1, Shazia Rafique 2 and Shin Min Kang 3,4,* 1 Division of Science and Technology, University of Education, Lahore 54000, Pakistan; [email protected] (M.M.); [email protected] (W.N.) 2 Center for Excellence in Molecular Biology, Punjab University Lahore, Lahore 53700, Pakistan; [email protected] 3 Department of Mathematics and Research Institute of Natural ...

General formulas are obtained for the vertex Padmakar-Ivan index (PIv) of tetrathiafulvalene (TTF) dendrimer, whereby TTF units we are employed as branching centers. The PIv index is a Wiener-Szeged-like index developed very recently. This topological index is defined as the summation of all sums of nu(e) and nv(e), over all edges of connected graph G.

We derive sharp lower bounds for the first and the second Zagreb indices (M1 and M2 respectively) for trees and chemical trees with the given number of pendent vertices and find optimal trees. M1 is minimized by a tree with all internal vertices having degree 4, while M2 is minimized by a tree where each “stem” vertex is incident to 3 or 4 pendent vertices and one internal vertex, while the res...

Several different topological indices (the carbon number, the Wiener index, the Balaban dis­ tance sum connectivity index, the Randic molecular identification number, and the Randic molecular connectivity indices °/, 2%, 3̂ p and 3/ c) were employed in the correlation of the velocity of ultrasonic sound in a variety of alkane and alcohol species. Using linear regression analyses and published so...

The first reformulated Zagreb index $EM_1(G)$ of a simple graph $G$ is defined as the sum of the terms $(d_u+d_v-2)^2$ over all edges $uv$ of $G .$ In this paper, the various upper and lower bounds for the first reformulated Zagreb index of a connected graph interms of other topological indices are obtained.

We give conditions on the numbers {φij} under which a vertexdegree-based topological index TI of the form