Scholar - SciOpen

Let $H$ be a simple, undirected, connected graph represented by its adjacency matrix $A$ . For a vertex $u \in V (H)$ , the generalized Gaussian subgraph centrality of $u$ in $H$ is $G S C (u, β) = \exp {(- β A^{2})}_{u u}$ , where $β > 0$ is the real number and represents the temperature. Furthermore, the generalized Gaussian Estrada index of $H$ is $G E E (H, β) = \sum_{i = 1}^{n} \exp (- β μ_{i}^{2}) = \sum_{u = 1}^{n} G S C (u, β)$ , where $μ_{1}, μ_{2}, \dots, μ_{n}$ are the eigenvalues of $A$ and $β > 0$ . This study presents new computational formulas for the $G S C (u, β)$ of graphs by employing an equitable partition and the star sets technique. We also investigated the influence of the parameter $β$ on the robustness of the formula through experiments. Additionally, we established some bounds for $G E E (H, β)$ .