On Some Bounds of Degree Based Topological Indices for Total Graphs

Abstract

In this paper, we discuss the concept of total graph and computed some topological indices. If $\Theta$ is a simple graph, then the elements of $\Theta$ are the vertices $\Theta_V$ and edges $\Theta_E$. For $ e=u\acute{u}\in \Theta_E$, the vertex $u$ and edge $e$, as well as $\acute{u}$ and $e$, are incident. We define the general harmonic $(GH)$ index and general sum connectivity $(GS)$ index for graph $\Theta$ regarding incident vertex-edge degrees as: $H_s^{\alpha}(\Theta)=\sum_{e\acute{u}}\big(\frac{2}{\aleph_{\acute{u}}+\aleph_{e}}\big)^{\alpha}$ and $\hat{\chi}_s^{\alpha}(\Theta)=\sum_{e\acute{u}}(\aleph_{\acute{u}}+\aleph_{e})^{\alpha}$, where $\alpha$ is any real number. In this article, we derive the closed formulas for a few standard graphs for $(GH)$ and $(GS)$ indices and then go on to calculate the lowest and the greatest general harmonic index, as well as the general sum-connectivity index, for various graphs that correspond to their total graphs.