A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs

Let $H$ be a simple, undirected, connected graph represented by its adjacency matrix $\mathit{A}$. For a vertex $u\in V(H)$, the generalized Gaussian subgraph centrality of $u$ in $H$ is $GSC(u,\beta )=\exp {\left(-\beta {\mathit{A}}^2\right)}_{uu}$, where $\beta >0$ is the real number and represents the temperature. Furthermore, the generalized Gaussian Estrada index of $H$ is $GEE(H,\beta )=\sum _{i=1}^n\exp \left(-\beta {\mu }_i^2\right)=\sum _{u=1}^{n}GSC(u,\beta )$, where ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ are the eigenvalues of $\mathit{A}$ and $\beta >0$. This study presents new computational formulas for the $GSC(u,\beta )$ of graphs by employing an equitable partition and the star sets technique. We also investigated the influence of the parameter $\beta$ on the robustness of the formula through experiments. Additionally, we established some bounds for $GEE(H,\beta )$.