THE GROUP INVERSE OF A MATRIX WITH A TREE GRAPH

Catral, Olesky and van den Driessche conjectured in 2008 a graph-theoretic formula for the group inverse of a singular real matrix whose digraph is a tree graph, with arcs in both directions along the edges of a tree. A maximum matching is a largest set of disjoint 2-cycles, and its weight is the product of the entries on its arcs. In the formula, the entry $(i, j)$ is a signed product of the entries along the path from $i$ to $j$, times the summed weight of the maximum matchings off the path, divided by that of the whole digraph. The entry is zero unless the path has odd length and removing its vertices lowers the term rank by their number. The authors and later papers verified it for paths, stars, brooms and one further class of trees. This note proves the formula for every tree. The proof reads the group inverse off the Laurent expansion of the resolvent at zero. In 2010, the same authors asked whether a formula of the same shape, summed over all paths of odd length from $i$ to $j$, holds for every matrix with a bipartite digraph whose largest sets of disjoint cycles have as many arcs as the rank and a nonzero signed sum of weights. The answer is negative, because a $4 \times 4$ matrix satisfies the hypotheses and violates the formula. The formula holds when, in addition, the term rank, the largest number of nonzero entries in distinct rows and columns, equals the rank.