Geodesic
On the characteristic polynomials of periodic graphs
Diskretnaya Matematika, Tome 16 (2004) no. 3, pp. 153-159.

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We consider periodic graphs of two types, chain and cyclic extension of an arbitrary graph. We give formulas which express the characteristic polynomials of such constructions in terms of the polynomials of the initial graphs, their subgraphs and the polynomials of a chain and a cycle. We present some applications of these results. 
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V. A. Kolmykov. On the characteristic polynomials of periodic graphs. Diskretnaya Matematika, Tome 16 (2004) no. 3, pp. 153-159. http://geodesic.mathdoc.fr/item/DM_2004_16_3_a8/

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[2] Schwenk A. J., Computing the characteristic polynomial of a graph, Springer-Verlag, Berlin - Heidelberg - New York:, 1974, 153–172 | MR

[3] Kolmykov V. A., “O summirovanii po putyam v spektralnoi teorii grafov”, Diskretnaya matematika, 13:4 (2001), 122–125 | MR | Zbl