Reduction procedures for calculating the determinant of the adjacency matrix of some graphs and the singularity of square planar grids

Let G be a graph without loops and multiple edges. If V(G) = (V[in1]), V2, …, vn, we define the adjacency matrix of G to be the n × n (0, I)-matrix A (G) = (aij), where aij = 1 if vivjϵ E (G) and aij = 0 otherwise. G is said to be singular if the matrix A(G) is singular. Reduction procedures which will decrease the amount of computation needed to obtain the determinant of the adjacency matrices of some graphs are introduced. One of these reduction procedures is used in proving the singularity of square planar grid Pn × Pn.