Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.

Two natural ways of doing it are:

1. By the degrees.
2. By the entries in a Perron eigenvector of the adjacency matrix.

These two methods coincide for regular graphs and for so-called harmonic graphs (defined as graphs in which the degree vector is an eigenvector) - which is all a tad trivial.

What is more interesting is that for many random graphs I've checked the two orderings coincide as well and I am able to show that they coincide for threshold graphs.

Have such graphs been studied?

I did find in the mathematical sociology literature some work on the question when the most central vertex w.r.t both rankings is the same but nothing for the whole vector.

(If such graphs haven't been named yet, I propose to call them tranquil, since tranquility is a lesser form of harmony).

Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.

Two natural ways of doing it are:

1. By the degrees.
2. By the entries in a Perron eigenvector of the adjacency matrix.

These two methods coincide for regular graphs and for so-called harmonic graphs (defined as graphs in which the degree vector is an eigenvector) - which is all a tad trivial.

What is more interesting is that for many random graphs I've checked the two orderings coincide as well and I am able to show that they coincide for threshold graphs.

Have such graphs been studied?

I did find in the mathematical sociology literature some work on the question when the most central vertex w.r.t both rankings is the same but nothing for the whole vector.

(If such graphs haven't been named yet, I propose to call them tranquil, since tranquility is a lesser form of harmony).

P.S. Method 2 is essentially what Google does in its PageRank algorithm.

Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.

Two natural ways of doing it are:

1. By the degrees.
2. By the entries in a Perron eigenvector of the adjacency matrix.

These two methods coincide for regular graphs and for so-called harmonic graphs (defined as graphs in which the degree vector is an eigenvector) - which is all a tad trivial.

What is more interesting is that for many random graphs I've checked the two orderings coincide as well and I am almost able to show that they coincide for threshold graphs (but not for split).

Have such graphs been studied?

I did find in the mathematical sociology literature some work on the question when the most central vertex w.r.t both rankings is the same but nothing for the whole vector.

(If such graphs haven't been named yet, I propose to call them tranquil, since tranquility is a lesser form of harmony).

Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.

Two natural ways of doing it are:

1. By the degrees.
2. By the entries in a Perron eigenvector of the adjacency matrix.

These two methods coincide for regular graphs and for so-called harmonic graphs (defined as graphs in which the degree vector is an eigenvector) - which is all a tad trivial.

What is more interesting is that for many random graphs I've checked the two orderings coincide as well and I am able to show that they coincide for threshold graphs.

Have such graphs been studied?

I did find in the mathematical sociology literature some work on the question when the most central vertex w.r.t both rankings is the same but nothing for the whole vector.

(If such graphs haven't been named yet, I propose to call them tranquil, since tranquility is a lesser form of harmony).