Linear mappings which preserve acyclicity properties of graphs and digraphs and applications to matrices

The issue of characterizing the linear transformations which map certain classes of square matrices into or onto themselves has been the theme of several papers recently written, e.g. [2-4 and 6]. The "into" problem is, in general, harder than the "onto" one, and it has been solved only under some additional hypothesis, namely, nonsingularity of the transformation or a somewhat weaker condition. Independently, there is a growing interest in learning the properties of acyclic matrices, i.e., square matrices whose (nondirected) graph contains no cycle except maybe for loops. These matrices, which are a natural generalization of tridiagonal matrices, are studied for example in [5, 7, 1] and the references there. Motivated by these two research directions, we investigate here the linear transformations which map the acyclic matrices into or onto themselves. In fact, we consider not only acyclic matrices but the general class d ~ k [ d ~ k] of all n x n matrices whose graph (digraph) contains no circuit (directed circuit) of length greater than or equal to k, k <~ n. Clearly, the set of all n x n acyclic matrices is the class d/~d~. The flavor of the discussion in this paper is different from that in [2-4 and 6] since, not surprisingly, the "into" problem here turns to be pure graph theoretic. In view of Lemma 6.3 we consider the equivalent problem of characterizing linear mappings on graphs (digraphs), as are defined in the next k k section, which map the set ~ [ ~ n ] of all graphs (digraphs) with n vertices which contain no circuit (directed circuit) of length greater than or equal to k into itself. Our results are proved under the assumption that the range of the mapping is wide enough to contain every possible edge (arc), except possibly for loops. This assumption is shown, by means of examples, to be necessary. We now describe our main results in more detail.