A categorification for the flow polynomial of graphs

M. Khovanov constructed a bigraded (co)homology group for links such that its graded Euler characteristic is equal to the Jones polynomial. L. Helme-Guizon and Y. Rong constructed a cohomology theory that categorifies the chromatic polynomial of graphs, i.e., the graded Euler characteristic of the cochain complex and the corresponding cohomology groups is the chromatic polynomial in [2, 3]. On the structures of the chromatic cohomology group, see [1, 4, 7]. E. F. Jasso- Hernandez and Y. Rong did the same for the Tutte polynomial of graphs in [5]. V. V. Vershinin and A. Y. Vesnin also did the same for the Yamada polynomial of graphs in [8]. K. Luse and Y. Rong did the same for the Penrose polynomial of plane graphs in [6]. The essential point of the construction is the state sum formula for polynomials. In this paper, for each graph G, we define bigraded cohomology groups, the Euler characteristic of which is a multiple of the flow polynomial of G. It is known that if a graph has a bridge, then its flow polynomial is zero. We show that this property is at the cohomology level