Geodesic
Two graphs with a common edge
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 3, pp. 497-507

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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.
Keywords: graph, adjacency matrix, determinant of graph, path, cycle 
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Badura, Lidia. Two graphs with a common edge. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 3, pp. 497-507. http://geodesic.mathdoc.fr/item/DMGT_2014_34_3_a2/