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, \beta) = \exp \left(- \beta \mathit{\boldsymbol{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^2_i \right) = \sum_{u = 1}^n GSC (u, \beta) $, where $ \mu_1, \mu_2, \ldots, \mu_n $ are the eigenvalues of $ \mathit{\boldsymbol{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) $.
Citation: Yang Yang, Yanyan Song, Haifeng Fan, Haiyan Qiao. A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs[J]. AIMS Mathematics, 2025, 10(2): 2279-2294. doi: 10.3934/math.2025106
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, \beta) = \exp \left(- \beta \mathit{\boldsymbol{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^2_i \right) = \sum_{u = 1}^n GSC (u, \beta) $, where $ \mu_1, \mu_2, \ldots, \mu_n $ are the eigenvalues of $ \mathit{\boldsymbol{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) $.
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Yang Yang, Yanyan Song, Haifeng Fan, Haiyan Qiao. A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs[J]. AIMS Mathematics, 2025, 10(2): 2279-2294. doi: 10.3934/math.2025106
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