Abstract
Topological indices have gained significant attention in the field of chemical graph theory. These indices offer quantitative measures that accurately represent the topology of molecular graphs, which are used to model chemical compounds. Generally, their physical properties are closely linked to the geometric structures of these compounds. In this paper, we introduce a new family of phenylene-quadrilateral networks that exhibit unique features. These studied topological structures can be seen as generalizations of the phenylenes. To analyze the generalized phenylene-quadrilateral networks, we propose a recursive method for calculating their Kirchhoff index and the number of spanning trees. This method is based on the relationship between the coefficients and roots of the characteristic polynomial.
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