Multipath cohomology of directed graphs

In this paper we introduce a new cohomology theory for (directed) graphs, which we call multipath cohomology. Our construction interpolates between the chromatic homology, introduced by L. Helme-Guizon and Y. Rong, and the homology for directed graphs introduced by P. Turner and E. Wagner. We prove that the multipath cohomology satisfies some functorial properties and we describe its connection with chromatic homology. We provide a number of sample computations showing that: the multipath cohomology is different from Turner-Wagner's and chromatic homologies, it does not vanish on trees, and, when evaluated at the coherent polygon, it recovers the Hochschild homology. We conclude by presenting some open questions and related problems.