Spanning tree centrality (STC) is a centrality measure based on the enumeration of spanning trees in a graph [2]. In a spanning tree, a node \(i\) can play two roles: either as a leaf, whose removal does not disconnect the tree, or as a cut-vertex, whose removal disconnects the tree.
The STC of node \(i\) is defined as
\[
c_{STC}(i) = t_G - d_i \, t_{G_i},
\]
where \(d_i\) is the degree of node \(i\), \(t_G\) is the total number of spanning trees in \(G\), and \(G_i\) is the subgraph obtained by removing node \(i\) from \(G\). The term \(d_i \, t_{G_i}\) counts the number of spanning trees in which node \(i\) is a leaf. Therefore, \(c_{STC}(i)\) measures the number of spanning trees in which node \(i\) acts as a cut-vertex, capturing its importance in maintaining the connectivity of the network.

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Qi, X., Fuller, E., Luo, R., & Zhang, C. Q. (2015). A novel centrality method for weighted networks based on the Kirchhoff polynomial. Pattern Recognition Letters, 58, 51-60. doi: 10.1016/j.patrec.2015.02.007.