Determinant of Graphs Joined by Two Edges

Supot Sookyang, Srichan Arworn, Boyko Gyurov

Abstract

In this paper we develop a technique to calculate the determinant of the adjacency matrix of a graph that is formed by joining two distinct simple graphs by two additional edges. The choice of the vertices at which the connection is established is an arbitrary and naturally the result is a function of the choice, so the technique is useful when the joined graphs are of special types - regular, strongly regular, complete graphs, wheel graphs, paths. In the second half of the paper we apply the technique to find the determinant of cycles joined by two edges and complete graphs joined by two edges.