Nowadays, with the development of the times, network structure analysis has become a hot issue in some fields. The eigenvalues of normalized Laplacian are very important for some network structure properties. Let Qn be octagonal-quadrilateral networks composed of n octagons and n squares and let Q2n be the strong prism of Qn. The strong product of a complete graph of order 2 and a complete graph of order G forms a strong prism of the graph G. In this paper, the decomposition theorem of the associated matrix is used to completely investigate the normalized Laplacian spectrum of Qn2. In addition, we establish exact formulas for the degree-Kirchhoff index and the number of spanning trees of Qn2.

Metal-Organic Networks (MON’s) is the central bone for the chemical compounds of the latest study for the energy department. The study of MON’s structure provides us numerous benefits in different fields related to chemical sciences, electrical and civil sectors. The MON’s structure is also used to restore different chemical compounds, especially those elements which can be used for the energy purpose such as hydrogen and carbon. Topological indices of the MON’s structure provide relationships between physical and chemical characteristics of the this compound such as melting points, boiling points, chemically stability, pressure, chemical reaction factors and many other basic properties. In this paper we calculate different topological indices based on first, second and third distances for two different metal-organic networks with expanding number of layers consisting on both organic ligands and metal vertices. A comparison between the calculated different kinds of the Topological Indices with the help of the numerical values and their graphical representation is also included.

The Sombor indices, a new category of degree-based topological molecular descriptors, havebeen widely investigated due to their excellent chemical applicability. This paper aims to establishSombor indices distributions in random polygonal chain networks and to achieve expressions of theexpected values and variances. The expected values and variances of the Sombor indices for polyonino,pentachain, polyphenyl, and cyclooctane chains are obtained. Since the end connection of a randomchain network follows a binomial distribution, the Sombor indices of any chain network follow the normaldistribution when the number of polygons connected by the chain, indicated by n, approaches infinity.

Many of the behaviors observed in actual systems are comparable to scale-free and small world structures in network research. In contrast to conventional hierarchical networks, the unusual fractal hierarchical network we created in this research has a pyramidal structure. The findings we get from this network are expanded to be applicable to arbitrary hierarchical networks. The average path length of unweighted and weighted hierarchical networks are the main topics of this paper. We demonstrate that, in the unweighted case, when the number of iterations z tends to infinity, the average path length is only related to the number of blocks of the hierarchical network. Additionally, in the weighted network, the average path length is related to the number of blocks r and the weighting factor w of the hierarchical network.

Spectrum graph theory not only facilitate comprehensively reflflect the topological structure and dynamic characteristics of networks, but also offer signifificant and noteworthy applications in theoretical chemistry, network science and other fifields. Let Ln (8, 4) represent a linear octagonal-quadrilateral network, consisting of n eight-member ring and n four-member ring. The M¨obius graph Qn(8, 4) is constructed by reverse identifying the opposite edges, whereas cylinder graph Q’n (8, 4) identififies the opposite edges by order. In this paper, the explicit formulas of Kirchhoffff indices and complexity of Qn(8, 4) and Q‘n (8, 4) are demonstrated by Laplacian characteristic polynomials according to decomposition theorem and Vieta’s theorem. In surprise, the Kirchhoffff index of Qn(8, 4)(Q’n (8, 4)) is approximately one-third half of its Wiener index as n → ∞.

In recent years, spectrum analysis and computation have developed rapidly in order to explore and characterize the properties of network sciences. Let Ln be obtained from the transformation of the graph L6,4,4 n , which obtained by attaching crossed two four-membered rings to the terminal of crossed phenylenes. Firstly, we study the (nomalized) Laplacian spectrum of Ln based on the decomposition theorem for the corresponding matrices. Secondly, we obtain the closed-term fomulas for the (multiplicative degree) Kirchhoff index and the number of spanning trees from the relationship between roots and coefficients in linear chain networks. Finally, we are surprised to find that the (multiplicative degree) Kirchhoff index of Ln is nearly to one quarter of its (Gutman) Wiener index when n tends to infinity.

Cycloarenes are a particular category of polycyclic aromatic hydrocarbons that have intrigued the experimental world for decades owing to the distinctiveness of their atomic and electrical configurations. They are suitable venues for investigating fundamental problems of aromaticity, particularly those involving the π-electron distribution in complex aromatic structures. Cycloarenes have recently attracted much attention due to their distribution as analogs for graphene pores. Kekulene is the member of this family that has been studied the most. For decades, its electrical structure has been a source of contention. It’s a doughnut-shaped chemical structure of circularly stacked benzene rings with interesting structural characteristics that lend themselves to experimental investigations like π-electron conjugation circuits. To predict their properties, topological characterization of such structures is required. This paper discusses two new series of big polycyclic compounds made by tessellating many kekulene doughnuts to make a hypothetical molecular belt with multiple cavities

For a long time, the structure and characteristics of benzene and other arenes have piqued researchers curiosity in quantum chemistry. The structural features of polycyclic aromatic compounds, like the fundamental molecular topology, have a strong influence on their chemical and biological properties. Quantitative structure-activity and property relationship (QSAR/QSPR) techniques for predicting characteristics of polycyclic aromatic compounds (PAC) and related graphs from chemical structures have been developed in this approach. To obtain degree-based topological indices, we have many polynomials. The neighbourhood M-polynomial is one of these polynomials, which is used to produce a number of topological indices based on neighborhood degree sum. In this study, we offer the exact analytical expressions of neighborhood M-polynomial and their corresponding topological indices for supercoronene (SC), cove-hexabenzocoronene (cHBC), and triangular-shaped discotic graphene (TDG) with hexabenzocorenene (HBC) as the base molecule. The findings could help with the development of physicochemical characteristic prediction.

Phenylenes network is applied in several fields of chemistry sciences due to its advantages compared to other several columnar networks, recently. This paper aims to introduce a kind of networks which obtained by a family of dicyclobutadieno derivative of linear phenylene chain Ln which is made up of n hexagons and (n+1) quadrangles. Let L2n be the strong prism of the dicyclobutadieno derivative of linear phenylenes Ln. By taking full advantage of the knowleges about the normalized Laplacian spectra, we induce the explicit expressions, with respect to the index n, of the multiplicative degree-Kirchhoff index and the number of spanning tree based on the graph L2n.

Cyclooctane is mainly used in the synthesis of cyclooctanone, cyclooctanol, caprolactam and octanoic acid. At the same time, it can also be used as an intermediate in organic synthesis and a chemical reagent. By discussing the resistance distance between any two points of cyclooctane derivative Tn(C8), some invariants about resistance distance are obtained, such as Kirchhoff index, multiplicative degree-Kirchhoff index, and additive degree-Kirchhoff index. Topological index can help scholars better understand some physical and chemical properties of compounds, and we obtain the closed expressions of valency-based topological indices for Tn(C8), such as ABC index, GA index, etc.

The Kirchhoff index and degree-Kirchhoff index have attracted extensive attentions due to their wide applications in physics and chemistry. These indices have been computed for many interesting graphs, such as linear polyomino chain, linear / Möbius / cylinder hexagonal chain, and linear octagonal chain. In the present paper, we consider Möbius octagonal chain (Mn) and cylinder octagonal chain (M’n). Explicit closed-form formulae of the Kirchhoff index and the number of spanning trees are obtained for Mn and M’n.