Stationary distribution for different types of graph

Question

This is a follow-up questions to posts:

Stationary distribution for directed graph

Stationary distribution for different types of graph

The definition of stationary distribution in wikipediaSteady-state analysis and limiting distributions

Are stationary distributions of graphs with every properties(for example directed or undirected, strongly connected or sparse, periodic or aperiodic)proportional with eigenvector corresponding to eigenvalue 1 or this property is only satisfied for strongly connected aperiodic graph?

If not, what is the difference for each case(I mean for example for Aperiodic or for periodic etc)?

Answer 1

The same Wikipedia article you link to gives the answer. This is true for any Markov chain with finite state space.

In other words, the stationary distribution π is a normalized (meaning that the sum of its entries is 1) left eigenvector of the transition matrix associated with the eigenvalue 1.