The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis o...

The atom-bond connectivity index is a useful topological index in studying the stability of alkanes and the strain energy of cycloalkanes. In this paper some inequalities for the atom-bond connectivity index of a series of graph operations are presented. We also prove our bounds are tight. As an application, the ABC indices of C4 nanotubes and nanotori are computed. © 2011 Elsevier B.V. All rig...

Abstract Topological indices of nanotubes are numerical descriptors that are derived from graph of chemical compounds. Such indices based on the distances in graph are widely used for establishing relationships between the structure of nanotubes and their physico-chemical properties. The Szeged index is obtained as a bond additive quantity where bond contributions are given as the product of th...

The forgotten topological index or F-index of a graph is defined as the sum of cubes of the degree of all the vertices of the graph. In this paper we study the F-index of four operations related to the lexicographic product on graphs which were introduced by Sarala et al. [D. Sarala, H. Deng, S.K. Ayyaswamya and S. Balachandrana, The Zagreb indices of graphs based on four new operations related...

In this paper, we present some inequalities for the Co-PI index involving the some topological indices, the number of vertices and edges, and the maximum degree. After that, we give a result for trees. In addition, we give some inequalities for the largest eigenvalue of the Co-PI matrix of G.

Muhammad Ajmal 1, Waqas Nazeer 2, Mobeen Munir 2, Shin Min Kang 3,4 and Young Chel Kwun 5,* 1 Department of Mathematics, Government Muhammdan Anglo Orintal College, Lahore 54000, Pakistan; [email protected] 2 Division of Science and Technology, University of Education, Lahore 54000, Pakistan; [email protected] (W.N.); [email protected] (M.M.) 3 Department of Mathematics and Research In...

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...

The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...

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 ...