Isoperimetric numbers of graph bundles
The isoperimetric number of a graph \(G\) is, roughly speaking, the minimum ratio \(|\partial X|/|X|\) of the size of a cocycle in \(G\) to the number of vertices in the smaller of the two parts of \(G-\partial X\). This invariant has been extensively studied in the recent years, see \textit{B. Mohar} [J. Comb. Theory, Ser. B 47, No. 3, 274-291 (1989; Zbl 0719.05042)]. The paper under review is devoted to upper and lower bounds of the isoperimetric number of a graph bundle. Graph bundle is a common generalization of a graph covering projection and the Cartesian product of graphs. The authors show that the isoperimetric number of any covering graph is not greater than that of the base graph. A more detailed evaluation is given in the case when the fibre is the complete graph. This extends a result of Mohar (loc. cit.) concerning the isoperimetric number of the Cartesian product of a graph and a complete graph.
scientific article; zbMATH DE number 934037