Sharath B, Gowtham H J
Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, 576104, India.
Sci Rep. 2025 Jul 5;15(1):24061. doi: 10.1038/s41598-025-09789-z.
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].
设[公式：见正文]是通过在子集[公式：见正文]中的每个顶点添加一个自环从简单图G得到的图。在本文中，我们将图[公式：见正文]的原子键连通性指数定义为[公式：见正文]，并将[公式：见正文]的原子键连通性能量定义为[公式：见正文]。我们得到了图[公式：见正文]的ABC谱半径的上界，以及关于m、n、Δ和δ的[公式：见正文]和[公式：见正文]的界。此外，我们计算了带有自环的完全图、鸡尾酒会图和皇冠图的ABC能量。我们还推导了带有自环的双星图的特征多项式。此外，我们探索了[公式：见正文]与各种物理化学性质之间的相关性，如沸点（BP）和摩尔折射（MR）。此外，我们建立了[公式：见正文]与特定指数之间的相关性，具体来说，是图[公式：见正文]的Sombor指数[公式：见正文]、图[公式：见正文]的第一 Zagreb 指数[公式：见正文]和图[公式：见正文]的 Randic 指数[公式：见正文]。
