Laplacian spectrum of rounded knot network and its applications

The Laplacian spectrum of a network encompasses the topology and structural characteristics of the network, as well as some dynamic characteristics, especially information related to random walks. In recent years, topological indices have become a research hotspot in the field of chemical graph theory, as these indices can accurately characterize the topological structure of molecular graphs for simulating compounds. The helix structure is the core structure of biomolecules and has received widespread attention from scientists in the chemical field. In this paper, the helix structure is introduced into the phenylene-quadrilateral networks, resulting in the generation of a rounded knot network. To analyze the network, based on the relationship between the coefficients and roots of the characteristic polynomial, we propose a recursive method to calculate its Kirchhoff index, the Mean First Passage Time (MFPT) and the number of spanning trees.