Let \(G_S\) be a graph derived from a simple graph G by adding a self-loop to each vertex in a subset \(S\subseteq V(G)\) . In this paper, we define the atom bond connectivity index of the graph \(G_S\) as \(ABC(G_S)\) and the atom bond connectivity energy of \(G_S\) as \(E_{ABC}(G_S)\) . We obtained upper bounds for the ABC spectral radius of the graph \(G_S\) as well as bounds for \(E_{ABC}(G_S)\) and \(ABC(G_S)\) in terms of m, n, \(\Delta\) and \(\delta\) . 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 \(ABC(G_S)\) and various physico-chemical properties, such as boiling point (BP) and molar refraction (MR). Furthermore, we established correlations between \(ABC(G_S)\) and specific indices, specifically the Sombor index of a graph \(G_S\) \((SO(G_S))\) , first Zagreb index of a graph \(G_S\) \((M_1(G_S))\) , and Randic index of a graph \(G_S\) \((R(G_S))\) .
