on the rank of the holomorphic solutions of pde associated to directed graphs

Let g be a directed graph with m vertices and n edges, i(b) the binomial ideal associated to the incidence matrix b of the graph g, and il the lattice ideal associated to the columns of the matrix b. also let bi be a submatrix of b after removing the ith column. in this paper it is determined that which minimal prime ideals of i(bi) are andean or toral. then we study the rank of the space of solutions of binomial d-module associated to i(bi) as a-graded ideal, where a is a matrix that, abi = 0. afterwards, we define a miniaml cellular cycle and prove that for computing this rank it is enough to consider these components of g. we introduce some bounds for the number of the vertices of the convex hull generated by the columns of the matrix a. finally an algorthim is introduced by which we can compute the volume of the convex hull corresponded to a cycles with k diagonals, so by theorem 2.1 the rank of (d / h a(i(b i); β) can be computed.