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I'm working on a graph problem somehow related to centrality measurement. Given an undirected, unweighted tree $T$ and a vertex $v$, let $D_i(v)$ be the set of vertices in $T$ that are i hops from $v$, $\epsilon(v)$ be eccentricity. then I propose a centrality measurement:

$$B(v)=\prod_{i=1}^{\epsilon(v)}|D_i(v)|!$$

Basically it calculates the number of ways to visit every other vertices starting from $v$ in BFS-style.

I'd like to study, if any, relationship between this centrality and closeness (aka distance) centrality. E.g. whether vertices with maximum above two centrality (so called centers) will coincide. So to better study property of them, I'm looking for some similar centrality measurement. So far I have known three centers will coincide in tree graph.

  • Closeness Center
  • Mass Center (Bohdan Zelinka, 1968)
  • Rumor Center (D. Shah and T. Zaman, 2011)

Could anyone provides some suggestion? Any thought is welcome and I'm glad to discuss with you all. Thanks a lot!

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