Several Results On The Determinant Of The Adjacency Matrix Of A Type Of Plane Bipartite Graphs

Let G be a simple graph and A(G)=(ai,j) the adjacency matrix of graph G, where ai,j=1if and only if (vi,Vj) is an edge of G and ai,j=0otherwise. Deift and Tomei (On the determinant of the adjacency matrix for a planar sublattice, J. Combin. Theory B35(1983)278-289.) proved that the determinant of the adjacency matrix for a finite subgraph G of Z×Z is0,1,-1, provided that G has no "holes"In the second chapter of this paper, we show that the determinant of A(G) of a plane graph G which has the property that every face-boundary is a cycle of length of the form4k(k=1,2,…), equals-1,0, or1, provided that the inner dual graph of G is a tree.It is well known that, if G is a simple graph with an even number of vertices and A(G) the adjacency matrix of graph G, then det A(G)=0(mod2) if and only if G has an even number of perfect matchings. Hence, if a bipartite graph G with n vertices satisfies the property det(A(G))=0or±1, by means of the above result, then:det(A(G))=0if G has an even number of perfect matchings and det(A(G))=±1otherwise.In the third chapter of this paper, based on the determinant of the adjacency matrix of the graph G, we classify some polygonal chains with length of the form4k(k=1,2,…). That is, if the determinant of the adjacency matrix of the polygonal chains G is0,1or-1, which polygonal chains G satisfying det(A(G))=0or det(A(G))=±1? We will answer completely this question.