On the topological indices of a generalized graph family used in number-theoretic graph realizations

Abstract

Abstract. In this paper, we study the topological properties of a generalized graph family, constructed by attaching loops to the vertices of a path graph. This family generalizes several graph structures that are used in the literature for realizing graph degree sequences based on number-theoretic sequences. We derive exact general formulas for various degree-based topological indices, including the first and second Zagreb indices, the Randic index, and the Harmonic index for this general family. As an application, we use these formulas to derive several new or known identities for Fibonacci and Lucas numbers by applying them to the known realizations of Fibonacci and Lucas graphs. Furthermore, we show that the ratio of these indices asymptotically converges to the cube of the Golden Ratio.