A Survey of directed graphs invariants

Abstract

In this paper, various kinds of invariants of directed graphs are summarized. In the first topic, the invariant w(G) for a directed graph G is introduced, which is primarily defined by S. Chen and X.M. Chen to solve a problem of weak connectedness of tensor product of two directed graphs. Further, we present our recent studies on the invariant w(G) in categorical view. In the second topic, Homology theory on directed graph is introduced, and we also cast on categorical view of the definition. The third topic mainly focuses on Laplacians on graphs, including traditional work and latest result of 1-laplacian by K.C.Chang. Finally, Zeta functions and Graded graphs are introduced, inclduing Bratteli-Vershik diagram, dual graded graphs and differential posets, with some applications in dynamic system.