Isoperimetric numbers of graph bundles

The main aim of this paper is to give some upper and lower bounds for the isoperimetric numbers of graph coverings or graph bundles, with exact values in some special cases. In addition, we show that the isoperimetric number of any covering graph is not greater than that of the base graph. Mohar's theorem for the isoperimetric number of the cartesian product of a graph and a complete graph can be extended to a more general case: The isoperimetric numberi(G × K2n) of the cartesian product of any graphG and a complete graphK2n on even vertices is the minimum of the isoperimetric numberi(G) andn, and it is also a sharp lower bound of the isoperimetric numbers of all graph bundles over the graphG with fiberK2n. Furthermore, ifn ≥ 2i(G) then the isoperimetric number of any graph bundle overG with fibreKn is equal to the isoperimetric numberi(G) ofG.

[1]  Bojan Mohar,et al.  The Maximum Genus of Graph Bundles , 1988, Eur. J. Comb..

[2]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[3]  J. Kwak,et al.  Isomorphism Classes of Graph Bundles , 1990, Canadian Journal of Mathematics.

[4]  David Avis,et al.  Balancing signed graphs , 1981, Discret. Appl. Math..

[5]  Bojan Mohar,et al.  Isoperimetric numbers of graphs , 1989, J. Comb. Theory, Ser. B.

[6]  N. Varopoulos,et al.  Isoperimetric inequalities and Markov chains , 1985 .

[7]  Peter L. Hammer.  Pseudo-Boolean remarks on balanced graphs , 1977 .

[8]  Michael Doob,et al.  Spectra of graphs , 1980 .

[9]  Tomaz Pisanski,et al.  Edge-colorability of graph bundles , 1983, J. Comb. Theory, Ser. B.

[10]  B. Mohar.  Isoperimetric inequalities, growth, and the spectrum of graphs , 1988 .

[11]  Peter J. Cameron,et al.  Signatures and signed switching classes , 1986, J. Comb. Theory, Ser. B.

[12]  J. Dodziuk.  Difference equations, isoperimetric inequality and transience of certain random walks , 1984 .

[13]  Jin Ho Kwak,et al.  Caracteristics polynomials of some grap bundlesII , 1992 .

[14]  Noga Alon,et al.  lambda1, Isoperimetric inequalities for graphs, and superconcentrators , 1985, J. Comb. Theory, Ser. B.

[15]  Béla Bollobás,et al.  An Isoperimetric Inequality on the Discrete Torus , 1990, SIAM J. Discret. Math..

[16]  Jonathan L. Gross,et al.  Generating all graph coverings by permutation voltage assignments , 1977, Discret. Math..