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**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.

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.) |

### Keywords:

characteristic polynomials; graph bundles; adjacency matrix; eigenvalues; automorphism group
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\textit{J. H. Kwak} and \textit{J. Lee}, Linear Multilinear Algebra 32, No. 1, 61--73 (1992; Zbl 0755.05078)

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### References:

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[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 |

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[6] | Hoffman K., Linear Algebra (1971) |

[7] | Kitamura T., Math. Japonica 35 pp 225– (1990) |

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[10] | Schwenk, A. J. 1974. ”Computing the characteristic polynomial of a graph”. Vol. 406, 153–172. New York: Springer-Verlag. |

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