Spectra, Hitting Times and Resistance Distances ofq- Subdivision Graphs

Abstract

Graph operations or products play an important role in complex networks. In this paper, we study the properties of q-subdivision graphs, which have been applied to model complex networks. For a simple connected graph G, its q-subdivision graph S q (G) is obtained from G through replacing every edge uv in G by q disjoint paths of length 2, with each path having u and v as its ends. We derive explicit formulas for many quantities of S q (G) in terms of those corresponding to G, including the eigenvalues and eigenvectors of normalized adjacency matrix, two-node hitting time, Kemeny constant, twonode resistance distance, Kirchhoff index, additive degree-Kirchhoff index, and multiplicative degree-Kirchhoff index. We also study the properties of the iterated q-subdivision graphs, based on which we obtain the closedform expressions for a family of hierarchical lattices, which has been used to describe scale-free fractal networks.

Spectral Analysis and its applications for a class of scale-free network based on the weighted m-clique annex operation

“…As a new and interdisciplinary science, complex network has attracted the attention of many scholars because of its application in various fields [Newman, 2003]. Modern science has confirmed that theory of graph spectra is an important tool for studying complex networks [Wu et al, 2019;Yi et al, 2018;Qi et al, 2018;Zeng & Zhang, 2021;Van Mieghem, 2010]. Many functions or dynamic properties on a complex network are difficult to directly respond to the topology of the network, but can be studied through the properties of the spectrum, such as: mixing time, Laplacian energy, hitting time, Kemeny constant, Multiplicative Degree-Kirchhoff index, the number of spanning trees, etc [Mehatari & Banerjee, 2015;Julaiti et al, 2013;Tetali, 1991;Chen & Zhang, 2007;Chang et al, 2014;Qi et al, 2015].…”

“…Furthermore, given that complex networks often have some special local structures, such as communities, motifs and cliques [Girvan & Newman, 2002;Milo et al, 2002;Tsourakakis, 2015], and graph operations and products can iteratively generate a large scale network with some special local structures from a small initial network, more and more attention has been paid to this generating mechanism of the complex network. In addition, due to the certainty of network operation rules, the properties of the spectrum are often studied systematically, which cannot be achieved in the real complex network [Wu et al, 2019;Yi et al, 2018;Qi et al, 2018;Zeng & Zhang, 2021]. At present, many network operations have been used to design and construct complex network models and to study the topological and dynamic properties on them, including: q−subdivision, planar triangulation, Cartesian product, hierarchical product, corona product, Kronecker product, etc.…”

Biharmonic distance of graphs

“…As another example, resistance distance, which originated in electrical circuit theory, has played a prominent role in circuit theory [11], chemistry [14,17], combinatorial matrix theory [3,21] and spectral graph theory [6]. Moreover, resistance distance has extensive applications ranging from quantifying biological structures [17], distributed control systems [4], network analysis [22], and power grid systems [19].…”