<![CDATA[This study aims to analyze the determination of the spectrum of barbell graph variations, where the variations are made by modifying the number of nodes on the bridge between complete graphs in a barbell structure. The spectrum contains the eigenvalues of the adjacency matrix of the barbell graph variations along with their multiplicities. The analysis is conducted manually using linear algebra approaches such as cofactor expansion, characteristic polynomial factorization, the rational root theorem, and Horner’s scheme. The results are then validated using Python programming. The findings of this study show that the longer and more complex the bridge connecting the two complete graphs, the greater the diversity of eigenvalues produced. The spectrum of the barbell graph B(n,1)B(n, 1)B(n,1) consists of the eigenvalues λ1,n−1,λ2,−1,λ3\lambda_1, n - 1, \lambda_2, -1, \lambda_3λ1,n−1,λ2,−1,λ3 with multiplicities 1,1,1,2n−3,11, 1, 1, 2n - 3, 11,1,1,2n−3,1. Furthermore, the spectrum of the barbell graph B(n,2)B(n, 2)B(n,2) consists of the eigenvalues λ1,λ2,λ3,λ4,−1,λ5,λ6\lambda_1, \lambda_2, \lambda_3, \lambda_4, -1, \lambda_5, \lambda_6λ1,λ2,λ3,λ4,−1,λ5,λ6 with multiplicities 1,1,1,1,2n−4,11, 1, 1, 1, 2n - 4, 11,1,1,1,2n−4,1, respectively. This research provides theoretical contributions regarding the relationship between complex graph structures and their spectral representations.]]>
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