Compact graphs and equitable partitions (Q677814)

Let \(\Gamma\) and \(S(A)\) be the sets of permutation and doubly-stochastic matrices, respectively, which commute with the adjacency matrix \(A\) of a graph \(G\). A graph \(G\) is called compact if every matrix from \(S(A)\) is a convex combination of matrices from \(\Gamma\). Graphs for which \(S(A)= \{I\}\) are characterized. It is proved that in compact regular graphs \(G\) any two vertices can be interchanged by an automorphism of \(G\).