Book Review: Introduction to the Theory of Benzenoid Hydrocarbons . by Ivan Gutman and Sven J. Cyvin; Springer-Verlag, Berlin, 1989, pp ix + 152, price DM 148

Many problems in real world, either natural or man-made, can be usefully represented by graphs or networks. Along with a complex topological structure, the weight is a vital factor in characterizing some properties of real networks. In this paper, we define a class of the weighted edge corona product networks. The generalized adjacency (resp., Laplacian and signless Laplacian) spectra with two different structures are determined. As applications, the number of spanning trees and Kirchhoff index of the weighted edge corona product networks are computed.

On the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks

Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as 􏽑(Γ) � 􏽑p∈V(Γ)[dΓ(p) 2] and 􏽑1(Γ) � 􏽑pq∈E(Γ)[dΓ(p) + dΓ(q)], respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as 􏽑2(Γ) � 􏽑pq∈E(Γ)[dΓ(p) × dΓ(q)]. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. )ey also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the first multiplicative Zagreb and second multiplicative Zagreb indices of the generalized F-sum graphs. At the end, some particular results as applications of the obtained results for alkane are also included.

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Multiplicative Zagreb Indices of Molecular Graphs

The Kirchhoff index and spanning trees of Möbius/cylinder octagonal chain

A molecular network can be characterized by several different ways, like a matrix, a polynomial, a drawing or a topological descriptor. A topological descriptor is a numeric quantity associated with a network or a graph that characterizes its whole structural properties. Analyzing and determining the topological indices and structural properties of a network or a graph have been a worthy studied topic in the field of chemistry, networks analysis, etc. In this paper, we consider several types of the generalized Sierpinski networks and investigate the explicit expressions of some well-known valency-based topological indices. Taking into account the other structural property of the generalized Sierpinski networks, the average degree is determined.

Valency-Based Topological Descriptors and Structural Property of the Generalized Sierpiński Networks

The computational complexity of the Hosoya index of a given graph is NP-Complete. Let RT(G) be the graph constructed from R(G) by a triangle instead of all vertices of the initial graph G. In this ...

The hosoya index of graphs formed by a fractal graph

On the number of spanning trees and normalized Laplacian of linear octagonal‐quadrilateral networks

Resistance distance is a novel distance function, also a new intrinsic graph metric, which makes some extensions of ordinary distance. Let $$O_n$$ be a linear crossed octagonal graph. Recently, Pan and Li (Int J Quantum Chem 118(24):e25787, 2018) derived the closed formulas for the Kirchhoff index, multiplicative degree-Kirchhoff index and the number of spanning trees of $$H_n$$. They pointed that it is interesting to give the explicit formulas for the Kirchhoff and multiplicative degree-Kirchhoff indices of $$O_n$$. Inspired by these, in this paper, two resistance distance-based graph invariants, namely, Kirchhoff and multiplicative degree-Kirchhoff indices are studied. We firstly determine formulas for the Laplacian (normalized Laplacian, resp.) spectrum of $$O_n$$. Further, the formulas for those two resistance distance-based graph invariants and spanning trees are given. More surprising, we find that the Kirchhoff (multiplicative degree-Kirchhoff, resp.) index is almost one quarter to Wiener (Gutman, resp.) index of a linear crossed octagonal graph.

Jing Zhao

Papers

Valency-Based Topological Descriptors and Structural Property of the Generalized Sierpiński Networks

The hosoya index of graphs formed by a fractal graph

On the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks

On the number of spanning trees and normalized Laplacian of linear octagonal‐quadrilateral networks

Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphs