Computing the degree of a vertex in the skeleton of acyclic Birkhoff polytopes

For a fixed tree T with n vertices the corresponding acyclic Birkhoff polytope Omega(n)(T) consists of doubly stochastic matrices having support in positions specified by T (matrices associated with T). The skeleton of Omega(n) (T) is the graph whose vertices are the permutation matrices associated with T and two vertices (permutation matrices) A and B are adjacent if and only if (E(G(A)) \ E(G(B))) U (E(G(B)) \ E(G(A))) is the edge set of a nontrivial path, where E(G(A)) and E(G(B)) are the edge sets of graphs associated with A and B, respectively. We present a formula to compute the degree of any vertex in the skeleton of Omega(n)(T). We also describe an algorithm for computing this number. In addition, we determine the maximum degree of a vertex in the skeleton of Omega(n)(T), for certain classes of trees, including paths and generalized stars where the branches have equal length. (C) 2015 Elsevier Inc. All rights reserved.