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.

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.