A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs - Yang Yang; Yanyan Song; Haifeng Fan; Haiyan Qiao (AIMS Mathematics)
Let $ H $ be a simple, undirected, connected graph represented by its adjacency matrix $ \mathit{\boldsymbol{A}} $. For a vertex $ u \in V(H) $, the generalized Gaussian subgraph centrality of $ u $ in $ H $ is $ GSC (u, β) = \exp \left(- β \mathit{\boldsymbol{A}}^2 \right)_{uu} $, where $ β > 0 $ is the real number and represents the temperature. Furthermore, the generalized Gaussian Estrada index of $ H $ is $ GEE(H, β) = \sum_{i = 1}^n \exp \left(- β μ^2_i \right) = \sum_{u = 1}^n GSC (u, β) $, where $ μ_1, μ_2, …, μ_n $ are the eigenvalues of $ \mathit{\boldsymbol{A}} $ and $ β > 0 $. This study presents new computational formulas for the $ GSC(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 $ GEE(H, β) $.