Eigenvalues of Antiadjacency Matrix of Directed Cyclic Dumbbell Graph

This paper explains the steps used to find the characteristic polynomial of the antiadjacency matrix of a cyclic dumbbell graph. The antiadjacency matrix of a graph is a matrix whose entries represent whether there is an edge that connects two vertices or not. The general form of the characteristic polynomial includes its eigenvalues. The antiadjacency matrix is obtained by using some theorems, the number of solutions of integer equations, quadratic formula, and polynomials factorization. Finally, results showed that the coefficients of the characteristic polynomial and its eigenvalues were dependent on the number of vertices of the cyclic dumbbell graph.