如何使用Java查找图的中心（顶点，该顶点与其他每个顶点相连，但边指向图的中心）

Question

how to Find the center of graph (vertex, that is connected with every other vertex, but edges are directed to the center of graph) with java. its very useful application for facebook like site.

Answer 1

Assume you have a graph with a set V of vertices:

V = { v1, v2, v3, ... , vn }

Now consider the extreme case where all vertices are connected to v2 and no other edges exist, ie the set of edges E given as tuples (from, to) is:

E = ( (v1, v2), (v3, v2), ... , (vn, v2) }

In this extreme case v2 clearly is the center of the graph as defined by you.

The connectivity matrix A looks like follows:

A = {
   from
to  v1, v2, v3, ..  vn
v1   0   0   0  ..   0    
v2   1   0   1  ..   1
v3   0   0   0  ..   0    
   :               :
vn   0   0   0  ..   0 }

Here v2 is clearly identified as the center of the graph by having a one in every position (excepting pos v2, ie itself) of its row of connectivity matrix A.

This would identify the center of graph even, when there are other edges in E. Note that there could be more than one center ...

A less strictly defined center of the graph could be found as the vertice with the most one entries in its row in the connectivity matrix.

You can avoid constructing the matrix A, when you have the set E and just count for each vertice how many times it appears in the to position of an edge tuple. The vertices with the maximum count are the losely defined centers of the graph, or the vertices with a count of n - 1 are your strictly defined centers.