On some interconnections between combinatorial optimization and extremal graph theory

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions

Finding minimal branchings with a given number of arcs

We describe an algorithm for finding a minimal s-branching (where s is a given number of its arcs) in a weighted digraph with an a symetric weight matrix. The algorithm uses the basic principles of the method (previously developed by J. Edmonds) for determining a minimal branching in the case when the number of its arcs is not specified in advance. Here we give a proof of the correctness for the described algorithm

A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph

We prove that the minimum value of the least eigenvalue of the signless Laplacian of a connected non-bipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices. © 2008 Elsevier Inc. All rights reserved.CEOCFCTFEDER/POCI 201