Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Vertex maps on graphs-trace theorems Chris Bernhardt | Bernhardt Fixed Point Theory and Applications 2011 2011 8 http content 2011 1 8 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Vertex maps on graphs-trace theorems Chris Bernhardt Correspondence cbernhardt@ Fairfield University Fairfield CT 06824 USA SpringerOpen0 Abstract The paper proves two theorems concerning the traces of Oriented Markov Matrices of vertex maps on graphs. These are then used to give a Sharkoksky-type result for maps that are homotopic to the identity and that flip at least one edge. 2000 Mathematics Subject Classification 37E15 37E25 Keywords Graphs Vertex maps Periodic orbits Sharkovsky s theorem Trace 1. Introduction A vertex map on a graph is a continuous map that permutes the vertices. Given a vertex map the periods of the periodic orbits can be computed giving a subset of the positive integers. One of the basic questions of combinatorial dynamics for vertex maps is to determine which subsets of the positive integers can be obtained in this way. Sharkovsky s theorem 1 is a well-known result that answers the question when the underlying graph is topologically an interval and the vertices all belong to the same periodic orbit. In 2 3 a Sharkovsky-type theorem was proved for trees. In the vertex map papers a standard method is to construct a matrix called the Oriented Markov Matrix. The entries along main diagonal of the matrix give information about periodic orbits. In particular the diagonal entries of the matrix raised to the nth power give information about the periodic orbits with period n. Thus the trace of powers of the matrix becomes important. In this paper two results concerning the trace of powers of the Oriented Markov Matrix are proved. The first shows that the trace is a homotopical invariant. The second shows how the trace can be calculated from the number of edges in the graph and the number of vertices that are not fixed by the vertex map. .
