Two graphs with a common edge

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.