Variable neighborhood search (VNS) is a recent metaheuristic for solving combinatorial and global optimization problems whose basic idea is systematic change of neighborhood within a local search. In this survey paper we present basic rules of VNS and some of its extensions. Moreover, applications are briefly summarized. They comprise heuristic solution of a variety of optimization problems, ways to accelerate exact algorithms and to analyze heuristic solution processes, as well as computer-assisted discovery of conjectures in graph theory.

Abstract. We study the algorithm that iteratively removes adjacent vertices from a simple, undirected graph until no edge remains. This algorithm is a well-known 2-approximation to three classical NP-hard optimization problems: MINIMUM VERTEX COVER, MINIMUM MAXIMAL MATCHING and MINIMUM EDGE DOMINATING SET. We show that the worst-case approximation factor of this simple method can be expressed in a finer way when assumptions on the density of the graph is made. For graphs with an average degree at least ɛn, called weakly ɛ-dense graphs, we show that the asymptotic approximation factor is min{2, 1/(1 − √ 1 − ɛ)}. For graphs with a minimum degree at least ɛn – strongly ɛ-dense graphs – we show that the asymptotic approximation factor is min{2, 1/ɛ}. These bounds are obtained through a careful analysis of the tight examples. 1

Abstract. The first attempt at automating mathematical conjecture-making appeared in the late-1950s. It was not until the mid-1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlying this program is the Principle of the Strongest Conjecture: make the strongest conjecture for which no counterexample is known. These two programs as well as other attempts to automate mathematical conjecture-making are surveyed—the success of a conjecture-making program, it is found, correlates strongly whether the program is designed to produce statements that are relevant to answering or advancing our mathematical questions. 1.

We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.

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journal homepage: www.elsevier.com/locate/laa On the reduced signless Laplacian spectrum of a degree maximal graph

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index and the diameter of a graph ∗

We consider a dynamic model of network formation where agents form and sever links based on the centrality of their potential partners. We show that the existence of capacity constrains in the amount of links an agent can maintain introduces a transition from dissortative to assortative networks. This effect can shed light on the distinction between technological and social networks as it gives a simple mechanism explaining how and why this transition occurs.

Recently Hansen and Vukičević [10] proved that the inequality M1/n ≤ M2/m, where M1 and M2 are the first and second Zagreb indices, holds for chemical graphs, and Vukičević and Graovac [17] proved that this also holds for trees. In both works is given a distinct counterexample for which this inequality is false in general. Here, we present some classes of graphs with prescribed degrees, that satisfy M1/n ≤ M2/m. Namely every graph G whose degrees of vertices are in the interval [c, c + ⌈ √ c ⌉] for some integer c, satisfies this inequality. In addition, we prove that for any ∆ ≥ 5, there is an infinite family of graphs of maximum degree ∆ such that the inequality is false. Moreover, an alternative and slightly shorter proof for trees is presented, as well as for unicyclic graphs.