Atom bond connectivity index for graph with self-loops and its application to structure property relationships in...

Let [Formula: see text] be a graph derived from a simple graph G by adding a self-loop to each vertex in a subset [Formula: see text]. In this paper, we define the atom bond connectivity index of the graph [Formula: see text] as [Formula: see text] and the atom bond connectivity energy of [Formula: see text] as [Formula: see text]. We obtained upper bounds for the ABC spectral radius of the graph [Formula: see text] as well as bounds for [Formula: see text] and [Formula: see text] in terms of m, n, Δ and δ. Additionally, we computed the ABC energy for complete graph, cocktail party graph and crown graph with self-loops. We also derived the characteristic polynomial of double star graph with self-loops. Furthermore, we explored the correlation between [Formula: see text] and various physico-chemical properties, such as boiling point (BP) and molar refraction (MR). Furthermore, we established correlations between [Formula: see text] and specific indices, specifically the Sombor index of a graph [Formula: see text] [Formula: see text], first Zagreb index of a graph [Formula: see text] [Formula: see text], and Randic index of a graph [Formula: see text] [Formula: see text].