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This paper explains the steps used to find the eigenvalues 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 2 vertices or not. The general form of the characteristic polynomial and the eigenvalues of the antiadjacency matrix are obtained by using some theorems, elementary row operation, the number of solutions of integer equations, quadratic formula, and polynomials factorization. Finally, it is found that the coefficients of the characteristic polynomial and its eigenvalues are dependent to the number of vertices of the cyclic dumbbell graph.

Antiadjacency Matrix; Characteristic Polynomials; Cyclic Dumbbell Graph; Eigenvalues