A Survey of directed graphs invariants

TL;DR

This survey comprehensively reviews invariants of directed graphs, including w(G), homology, Laplacians, and zeta functions, highlighting recent developments and categorical perspectives.

Contribution

It consolidates various invariants of directed graphs, introduces categorical views, and discusses recent advances in homology, Laplacians, and applications in dynamic systems.

Findings

Categorical perspective on graph invariants

Recent results on 1-Laplacian by K.C. Chang

Applications of zeta functions and graded graphs in dynamics

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.