Linear operators that strongly preserve graphical properties of matrices

Abstract

An operator on the set M of n × n matrices strongly preserves a subset F if it maps F into F and M \\ F into M \\ F . The operator semigroup of F is the semigroup of linear operators strongly preserving F . We show that all the n × n matrix-families which are determined by the directed graphs of their members and satisfy a short list of conditions, have the same operator semigroup Σ, and we determine the generators of Σ. Among those matrix-families are: the irreducible matrices; the matrices whose directed graphs have maximum cycle length l ⩾ k for fixed k ⩾ 4; and the matrices whose directed graphs have a path of length at least l ⩾ k for fixed k ⩾ 3. Similar results are obtained for matrix-families determined by the undirected graphs of their members.