Valency-Based Topological Properties of Linear Hexagonal Chain and Hammer-Like Benzenoid

Topological indices are quantitative measurements that describe a molecule’s topology and are quantified from the molecule’s graphical representation. The significance of topological indices is linked to their use in QSPR/QSAR modelling as descriptors. Mathematical associations between a particular molecular or biological activity and one or several biochemical and/or molecular structural features are QSPRs (quantitative structure-property relationships) and QSARs (quantitative structure-activity relationships). In this paper, we give explicit expressions of two recently defined novel ev-degree- and ve-degree-based topological indices of two classes of benzenoid, namely, linear hexagonal chain and hammer-like benzenoid.

Omega, Sadhana, and PI Polynomials of Quasi-Hexagonal Benzenoid Chain

Counting polynomials are important graph invariants whose coefficients and exponents are related to different properties of chemical graphs. Three closely related polynomials, i.e., Omega, Sadhana, and PI polynomials, dependent upon the equidistant edges and nonequidistant edges of graphs, are studied for quasi-hexagonal benzenoid chains. Analytical closed expressions for these polynomials are derived. Moreover, relation between Padmakar–Ivan (PI) index of quasi-hexagonal chain and that of corresponding linear chain is also established.

Sharp Bounds for the Second-Order General Connectivity Index of Hexagonal Chains

Given H, a hexagonal chain, we determine an expression of its second-order general connectivity index, denoted by χ2α(H), in terms of inlet features of H. Moreover, by applying the method in integer programming theory, we completely determine the extremal chains with the minimal or maximal χ2α(H) over the set of hexagonal chains.

Applications on Hyper-Zagreb Index of Generalized Hierarchical Product Graphs

Let G be a connected graph. The Hyper-Zagreb index of a connected graph G is defined as HM(G) = Σuv∈EG [dG(u)+dG(v)]2, where dG(v) is the degree of the vertex v in G. In this paper, the Hyper-Zagreb Gindex of the generalized hierarchical, Cartesian, cluster, corona products and four new sums of graphs according to some invariants of the factors are computed, respectively. As applications, we present explicit formulas for the HM index of the linear phenylene Fn, the C4 nanotorus Cm□Cn, the C4 nanotubes Pm□Cn, the l-dimensional hypercubes Ql , the zig-zag polyhex nanotube TUHC6[2n, 2], the hexagonal chain ln, the regular dicentric dendrimer DDp,r and so forth.

Second Atom-Bond Connectivity Index of Special Chemical Molecular Structures

In theoretical chemistry, the second atom-bond connectivity index was introduced to measure the stability of alkanes and the strain energy of cycloalkanes. In this paper, we determine the second atom-bond connectivity index of unilateral polyomino chain and unilateral hexagonal chain. Furthermore, the secondABCindices of V-phenylenic nanotubes and nanotori are presented.

Kirchhoff Index of Cyclopolyacenes

The resistance distance between two vertices of a connected graph G is computed as the effective resistance between them in the corresponding network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices. In this paper, following the method of Y. J. Yang and H. P. Zhang in the proof of the Kirchhoff index of the linear hexagonal chain, we obtain the Kirchhoff index of cyclopolyacenes, denoted by HRn, in terms of its Laplacian spectrum. We show that the Kirchhoff index of HRnis approximately one third of its Wiener index.