About properties of adjacency matrix when a graph is planar 1- Is there any specific properties for adjacency matrix when a graph is planar?
2- Is there any thing special for computing the permanent of adjacency matrix when a graph is planar?
Computing determinant and permanent of planar graphs are as hard as computing them in general graphs. They are complete for GapL and #P respectively. See this paper by Datta, Kulkarni, Limaye, Mahajan for more details.