On Compact Graphs

Let G be a finite simple graph with adjacency matrix A, and let P..(A) be the convex closure of the set of all permutation matrices commuting with A. G is said to be compact if every doubly stochastic matrix which commutes with A is in P.(A). In this paper, we characterize 3.regular compact graphs and prove that if G is a connected regular compact graph, G . v is also compact, and give a family of almost regular compact connected graphs.