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

## TL;DR

This paper introduces a new atom bond connectivity index for graphs with self-loops and explores its relationship with physico-chemical properties of anticancer drugs.

## Contribution

The paper defines and analyzes a new atom bond connectivity index for graphs with self-loops and applies it to anticancer drug structure-property relationships.

## Key findings

- Upper bounds for the ABC spectral radius and energy of graphs with self-loops were derived.
- Correlations between the ABC index and physico-chemical properties like boiling point and molar refraction were established.
- The ABC index was correlated with other graph indices such as the Sombor, Zagreb, and Randic indices.

## Abstract

Let \documentclass[12pt]{minimal}
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\begin{document}$$G_S$$\end{document} be a graph derived from a simple graph G by adding a self-loop to each vertex in a subset \documentclass[12pt]{minimal}
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\begin{document}$$S\subseteq V(G)$$\end{document}. In this paper, we define the atom bond connectivity index of the graph \documentclass[12pt]{minimal}
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\begin{document}$$G_S$$\end{document} as \documentclass[12pt]{minimal}
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\begin{document}$$ABC(G_S)$$\end{document} and the atom bond connectivity energy of \documentclass[12pt]{minimal}
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\begin{document}$$G_S$$\end{document} as \documentclass[12pt]{minimal}
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\begin{document}$$E_{ABC}(G_S)$$\end{document}. We obtained upper bounds for the ABC spectral radius of the graph \documentclass[12pt]{minimal}
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\begin{document}$$G_S$$\end{document} as well as bounds for \documentclass[12pt]{minimal}
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\begin{document}$$E_{ABC}(G_S)$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$ABC(G_S)$$\end{document} in terms of m, n, \documentclass[12pt]{minimal}
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\begin{document}$$\Delta$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$\delta$$\end{document}. 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 \documentclass[12pt]{minimal}
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\begin{document}$$ABC(G_S)$$\end{document} and various physico-chemical properties, such as boiling point (BP) and molar refraction (MR). Furthermore, we established correlations between \documentclass[12pt]{minimal}
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\begin{document}$$ABC(G_S)$$\end{document} and specific indices, specifically the Sombor index of a graph \documentclass[12pt]{minimal}
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\begin{document}$$G_S$$\end{document}
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\begin{document}$$(SO(G_S))$$\end{document}, first Zagreb index of a graph \documentclass[12pt]{minimal}
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\begin{document}$$G_S$$\end{document}
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\begin{document}$$(M_1(G_S))$$\end{document}, and Randic index of a graph \documentclass[12pt]{minimal}
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\begin{document}$$G_S$$\end{document}
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\begin{document}$$(R(G_S))$$\end{document}.