Graph Center

The graph center of a graph is the set of vertices of graph eccentricity equal to the graph radius (i.e., the set of central points). In the above illustration, center nodes are shown in red.

For a connected graph, the center may be computed in the Wolfram Language with the command GraphCenter[g].

Here, graph distances between distinct connected components are taken to be infinite. Consequently, every graph vertex of a disconnected graph has infinite graph eccentricity and is central.

Using this convention, the following table gives the number of -node simple unlabeled graphs having center nodes.

	OEIS	, 2, ...
1	A052437	1, 0, 1, 2, 8, 29, 180, ...
2	A052438	0, 2, 0, 2, 4, 19, 84, ...
3	A052439	0, 0, 3, 0, 4, 18, 119, ...
4	A052440	0, 0, 0, 7, 0, 18, 118, ...
5	A052441	0, 0, 0, 0, 18, 0, 129, ...
6		0, 0, 0, 0, 0, 72, 0, ...
7		0, 0, 0, 0, 0, 0, 414, ...