Is my understanding of the concept of centres of directed, weighted graphs correct?

Question

I'm currently working on a problem that requires me to find the centre of a directed, weighted graph. I am seeking to ensure that my understanding of some related concepts is correct.

For instance, let's say we have some sets of nodes represented as links:

/wiki/Flow_network
/wiki/Braess%27_paradox

/wiki/Flow_network
/wiki/Circulation_problem

/wiki/Braess%27_paradox
/wiki/new

/wiki/new
/wiki/Braess%27_paradox

Each set has two nodes (links), where the first node is the "source" node and has a directed edge to the second node.

As I understand it, each of the nodes have the following eccentricities:

ecc(FN) = 2
ecc(CP) = 0
ecc(BP) = 1
ecc(new) = 1

The radius of the graph would be 0, since this is the smallest eccentricity.

And since the centre of a graph is the set of nodes which have eccentricity = radius, the centre of this directed, weighted graph would be CP?

One reason I'm seeking to ensure that my understanding is correct is because this "centre" seems weird when one draws the graph in question.

Have I understood this correctly?

Answer 1

Before reading, note that I'm not a mathematician, and I'm simply trying to attempt to answer this with implementation in mind. The definition of the center of a graph is indeed the set of all vertices of minimum eccentricity. The problem is that this is usually a concept used on undirected graphs. If your graph is undirected, you won't run into issues like the one you experienced here where your vertex of least eccentricity doesn't connect to any other vertices. By definition, you are correct that this is the "Center" of the graph. However, this would obviously not be the center if the graph were undirected and is probably useless to you in anything other than the theoretical context. If you're simply trying to find the theoretical center of the graph, this is probably your answer, at least if you follow the definitions of of eccentricity, radius, and center found here: https://en.wikipedia.org/wiki/Distance_(graph_theory). If you're trying to find something more to the effect of the center of an undirected graph, where the vertex returned is the least far away from all other vertices, maybe try finding the vertex with lowest eccentricity that has a path to all or most other nodes, or maybe set the eccentricity of a vertex to infinity if it doesn't connect to any other nodes. Either of those suggestions will likely get you more useful results. If you want a more theoretical view, head over to the math stackexchange: https://math.stackexchange.com/.