Eigenvalues of large even annulenes in terms ofK 3,S 4 andS 5

Squares of the adjacency matrices of bipartite cycles (C_v) can be block-factored into matrices which correspond to vertex-weighted complete graphs forv = 6, vertex-weighted strongly regular graphs forv = 8 and 10, and vertex-weighted metrically regular graphs forv > 10. Using this fact and some properties of strongly and metrically regular graphs, it is shown that eigenvalues of large bipartite C_v graphs (i.e. large even annulenes) can be expressed by the general formula ± (2 ± (2 ± (... ± (2 +r_p)) ...), wherev = 2ⁿ ×p,n is the number of surd () signs required andp = 3, 4 and 5. Here,r₃,r₄, andr₅, are the eigenvalues of the complete graphK₃ and the strongly regular graphsS₄ andS₅ respectively. The procedure does not require construction of characteristic polynomials for the determination of eigenvalues, and brings out a common topological origin for the two-fold degeneracies observed in the eigenvalue spectra of all even cycles and many odd cycles.