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

Kwak, Jin Ho; Lee, Jaeun

Linear Multilinear Algebra 32, No. 1, 61-73 (1992).

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.)

Keywords:

characteristic polynomials; graph bundles; adjacency matrix; eigenvalues; automorphism group

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.