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