On spectral bounds for cutsets

Let Γ be a simple and connected graph. A k-vertex separator [k-edge separator] is a subset of vertices [edges] whose deletion separates the vertex [edge] set of Γ into two parts of equal cardinality, that are at distance greater than k in Γ. Here we investigate the relation between the cardinality of these cutsets and the laplacian spectrum of Γ. As a consequence of the study, we obtain the well-known lower bounds for the bandwidth and the bipartition width of a graph.