## Characteristic polynomials of some graph bundles. II.(English)Zbl 0755.05078

Summary: [Part I, cf. Can. J. Math. 42, No. 4, 747-761 (1990).]
The characteristic polynomial of a graph $$G$$ is that of its adjacency matrix, and its eigenvalues are those of its adjacency matrix. Recently, Y. Chae, J. H. Kwak and J. Lee showed a relation between the characteristic polynomial of a graph $$G$$ and those of graph bundles over $$G$$. In particular, the characteristic polynomial of $$G$$ is a divisor of those of its covering graphs. They also gave the complete computation of the characteristic polynomials of $$K_ 2$$ (or $$\overline K_ 2)$$- bundles over a graph. In this paper, we compute the characteristic polynomial of a graph bundle when its voltages lie in an abelian subgroup of the full automorphism group of the fibre; in particular, the automorphism group of the fibre is abelian. Some applications to path- or cycle-bundles are also discussed.

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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### References:

 [1] Biggs N., Algebraic Graph Theory (1974) · Zbl 0284.05101 [2] Chae Y., Characteristic polynomials of some graph bundles (1974) [3] Cvetkovié D. M., Spectra of Graphs (1979) [4] Gross J. L., Topological Graph Theory (1987) · Zbl 0621.05013 [5] DOI: 10.1016/0012-365X(77)90131-5 · Zbl 0375.55001 [6] Hoffman K., Linear Algebra (1971) [7] Kitamura T., Math. Japonica 35 pp 225– (1990) [8] DOI: 10.4153/CJM-1990-039-3 · Zbl 0739.05042 [9] Mohar B., Europ. J. Combinatorics 9 pp 215– (1988) [10] Schwenk, A. J. 1974. ”Computing the characteristic polynomial of a graph”. Vol. 406, 153–172. New York: Springer-Verlag.
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