Sharp Bounds of Kulli–Basava Indices in Generalized Form for k-Generalized Quasi Trees

Abstract

Molecular descriptors are a basic tool in the spectral graph, molecular chemistry, and various other fields of mathematics and chemistry. Kulli–Basava $\left(\mathbb{K}\mathfrak{B}\right)$ indices were initiated for chemical applications of various substances in chemistry. For simple graph $G$ , $\mathbb{K}\mathfrak{B}$ indices in generalized forms are $\mathbb{K}{\mathfrak{B}}_1^{\mathrm{\varrho }}\left(G\right)={{\displaystyle \sum }}_{gh\in E\left(G\right)}{\left({\mathbb{S}}_e\left(g\right)+{\mathbb{S}}_e\left(h\right)\right)}^{\mathrm{\varrho }}$ and $\mathbb{K}{\mathfrak{B}}_2^{\mathrm{\varrho }}\left(G\right)={{\displaystyle \sum }}_{gh\in E\left(G\right)}{\left({\mathbb{S}}_e\left(g\right).{\mathbb{S}}_e\left(h\right)\right)}^{\mathrm{\varrho }}$ , where ${\mathbb{S}}_e\left(g\right)={{\displaystyle \sum }}_{e\in N_e\left(g\right)}{d}_G\left(e\right)$ , and for edge $e=\left\{g,h\right\}$ , the degree is $d_G\left(e\right)=d_G\left(g\right)+d_G\left(h\right)-2$ and $\mathrm{\varrho }\ne 0$ is any real number. The graph $G$ is said to be a $k-\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d}$ quasi tree if for the vertex set $U_k\subset G$ having $\left|U_k\right|=k$ , $G-U_k$ is a tree and for $U_{k-1}\subset V\left(G\right)$ having $\left|U_{k-1}\right|=k-1$ , $G-U_{k-1}$ is not a tree. In this research work, we have successfully investigated sharp bounds of generalized $\mathbb{K}\mathfrak{B}$ indices for k-generalized quasi trees where $\mathrm{\varrho }\ge 1$ . Chemical applications of the generalized form are also studied for alkane isomers with scatter diagrams and residuals.