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.
Fixed Point Theory and Applications
Bernhardt, Chris, "Vertex maps on graphs-trace theorems" (2011). Math & Computer Science Faculty Publications. 22.
Bernhardt, Chris. (2011). Vertex maps on graphs-trace theorems. Fixed Point Theory and Applications - A SpringerOpen Journal 2011:8