Two Graphs with a Common Edge

摘要

Let G = G_(1) ∪ G_(2) be the sum of two simple graphs G_(1),G_(2) having a common edge or G = G_(1) ∪ e_(1) ∪ e_(2) ∪ G_(2) be the sum of two simple disjoint graphs G_(1),G_(2) connected by two edges e_(1) and e_(2) which form a cycle C_(4) inside G. We give a method of computing the determinant det A(G) of the adjacency matrix of G by reducing the calculation of the determinant to certain subgraphs of G_(1) and G_(2). To show the scope and effectiveness of our method we give some examples.