The benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem. The performance of this new method, called extremal optimization (EO), is compared with simulated annealing (SA) in extensive numerical simulations. While generally a complex (NP-hard) problem, the optimization of the graph partitions is particularly difficult...

Ryser’s Conjecture states that any r-partite r-uniform hypergraph has a vertex cover of size at most r− 1 times the size of the largest matching. For r = 2, the conjecture is simply König’s Theorem. It has also been proven for r = 3 by Aharoni using a beautiful topological argument. This paper is the first part of the proof of a characterization of those 3-uniform hypergraphs for which Aharoni’...

We determine the minimum sum–connectivity index of bicyclic graphs with n vertices and matching number m, where 2 ≤ m ≤ ⌊n2 ⌋, the minimum and the second minimum, as well as the maximum and the second maximum sum–connectivity indices of bicyclic graphs with n ≥ 5 vertices. The extremal graphs are characterized. MSC 2000: 05C90; 05C35; 05C07

My research interests lie in spectral and extremal graph theory, as well as in the analysis of complex networks. A primary goal is to deduce fundamental structural graph properties from the spectrum of the graph or a few easily-computable invariants. In order to gain control over these invariants, extremal graph theory aims to tightly bound them and characterize the graphs achieving these bound...

Edge-independent random graphs are a model of random graphs in which each potential edge appears independently with an individual probability. Based on the relative entropy method, we determine the upper and lower bounds for the extremal vertex degrees using the edge probability matrix and its largest eigenvalue. Moreover, an application to random graphs with given expected degree sequences is ...

In this paper, we investigate how the Wiener index of unicyclic graphs varies with graph operations. These results are used to present a sharp lower bound for the Wiener index of unicyclic graphs of order n with girth and the matching number β ≥ 3g 2 . Moreover, we characterize all extremal graphs which attain the lower bound.

A function f bounds graphs from above if there exists an infinite family of graph G, such that if G ∈ G then f(|VG|) = |EG| and for all nonempty subgraphs H of G we have that F (|VH |) ≥ |EH |. This paper considers the question: which functions bound graphs? keywords graphs, extremal graph theory AMS subject classification 05C99

Extremal optimization is a new general-purpose method for approximating solutions to hard optimization problems. We study the method in detail by way of the computationally hard (NP-hard) graph partitioning problem. We discuss the scaling behavior of extremal optimization, focusing on the convergence of the average run as a function of run time and system size. The method has a single free para...