Maximum Signless Laplacian Estrada Index of Tetracyclic Graphs

Abstract

In this research paper, we explore the Atom-Bond Sum-Connectivity (ABS) index for unicyclic graphs. The ABS index for a graph $G$ is defined by the formula: $$ABS(G) = {\sum\limits_{\omega \gamma \in E(G)}} \sqrt{\dfrac{{d_\omega}+{d_\gamma}-2}{{d_\omega}+{d_\gamma}}},$$ where ${d_\omega}$ represents the degree of vertex $\omega$ in the graph $G$. Our study aims to identify the unicyclic graph with the maximum ABS index among those with a given order $\rho$ and diameter $\Omega$. Furthermore, we provide a characterization of all unicyclic graphs that achieve this maximum ABS index.