Atom bond connectivity index for graph with self-loops and its application to structure property relationships in anticancer drugs
B. Sharath, H. J. Gowtham
Abstract 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 $$G_S$$ $$(R(G_S))$$ .
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:Scientific Reports
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