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{\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 $ \...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2025-02-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2025106 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850216295505592320 |
|---|---|
| author | Yang Yang Yanyan Song Haifeng Fan Haiyan Qiao |
| author_facet | Yang Yang Yanyan Song Haifeng Fan Haiyan Qiao |
| author_sort | Yang Yang |
| collection | DOAJ |
| description | 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) $. |
| format | Article |
| id | doaj-art-ba513aa5cbfb495c9caee6988d25516c |
| institution | OA Journals |
| issn | 2473-6988 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-ba513aa5cbfb495c9caee6988d25516c2025-08-20T02:08:20ZengAIMS PressAIMS Mathematics2473-69882025-02-011022279229410.3934/math.2025106A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphsYang Yang0Yanyan Song1Haifeng Fan2Haiyan Qiao3College of Artificial Intelligence, Tianjin University of Science and Technology, Tianjin 300457, ChinaSchool of Mathematics and Statistics, Qinghai Normal University, Xining 810008, ChinaCollege of Artificial Intelligence, Tianjin University of Science and Technology, Tianjin 300457, ChinaSchool of Information and Electrical Engineering, Hebei University of Engineering, Handan 056038, ChinaLet $ 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) $.https://www.aimspress.com/article/doi/10.3934/math.2025106generalized gaussian estrada indexgeneralized gaussian subgraph centralitystatistical mechanicsadjacency matrixequitable partition |
| spellingShingle | Yang Yang Yanyan Song Haifeng Fan Haiyan Qiao A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs AIMS Mathematics generalized gaussian estrada index generalized gaussian subgraph centrality statistical mechanics adjacency matrix equitable partition |
| title | A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs |
| title_full | A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs |
| title_fullStr | A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs |
| title_full_unstemmed | A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs |
| title_short | A note on the generalized Gaussian Estrada index and Gaussian subgraph centrality of graphs |
| title_sort | note on the generalized gaussian estrada index and gaussian subgraph centrality of graphs |
| topic | generalized gaussian estrada index generalized gaussian subgraph centrality statistical mechanics adjacency matrix equitable partition |
| url | https://www.aimspress.com/article/doi/10.3934/math.2025106 |
| work_keys_str_mv | AT yangyang anoteonthegeneralizedgaussianestradaindexandgaussiansubgraphcentralityofgraphs AT yanyansong anoteonthegeneralizedgaussianestradaindexandgaussiansubgraphcentralityofgraphs AT haifengfan anoteonthegeneralizedgaussianestradaindexandgaussiansubgraphcentralityofgraphs AT haiyanqiao anoteonthegeneralizedgaussianestradaindexandgaussiansubgraphcentralityofgraphs AT yangyang noteonthegeneralizedgaussianestradaindexandgaussiansubgraphcentralityofgraphs AT yanyansong noteonthegeneralizedgaussianestradaindexandgaussiansubgraphcentralityofgraphs AT haifengfan noteonthegeneralizedgaussianestradaindexandgaussiansubgraphcentralityofgraphs AT haiyanqiao noteonthegeneralizedgaussianestradaindexandgaussiansubgraphcentralityofgraphs |