In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which is an analog of the Brualdi-Hoffman conjecture for general graphs, and prove the conjecture in some special cases.

An elegant and general way to apply graph partitioning algorithms to hypergraphs would be to model hypergraphs by graphs and apply the graph algorithms to these models Of course such models have to simulate the given hypergraphs with respect to their cut properties An edge weighted graph V E is a cut model for an edge weighted hypergraph V H if the weight of the edges cut by any bipartition of ...

Given a graph G and a bipartition of its vertices, the edge-ratio is the minimum for both classes so defined of their number of internal edges divided by their number of cut edges. We prove that maximizing edgeratio is NP-complete.

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. In the paper, we characterize the graphs with minimal energy in the class of bipartite unicyclic graphs of a given (p, q)–bipartition, where q ≥ p ≥ 2.

A hereditary class P is called finitely generated if the set of all minimal forbidden induced subgraphs for P is finite. For a pair of hereditary classes P and Q, we define a hereditary class P ∗ Q of all graphs G which have a partition A ∪ B = V (G) such that G(A) ∈ P and G(B) ∈ Q, where G(X) denotes the subgraph of G induced by X ⊆ V (G). We investigate the problem of recognizing finitely gen...

Alon, Bollobás, Krivelevich and Sudakov [1] proved that every graph with a large cut has a bipartition in which each vertex class contains correspondingly few edges. We prove an analogous result for partitions into k ≥ 3 classes; along the way we prove a result for biased bipartitions.