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53
Variable Neighborhood Search
, 1997
"... 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 a ..."
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Cited by 355 (26 self)
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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 computerassisted discovery of conjectures in graph theory.
Eigenvalue bounds for the signless Laplacian
 PUBL. INST. MATH. (BEOGRAD
, 2007
"... We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the ..."
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Cited by 22 (0 self)
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We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computergenerated conjectures.
Decreasing the spectral radius of a graph by link removals
, 2011
"... The decrease of the spectral radius, an important characterizer of network dynamics, by removing links is investigated. The minimization of the spectral radius by removing m links is shown to be an NPcomplete problem, which suggests to consider heuristic strategies. Several greedy strategies are co ..."
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Cited by 13 (6 self)
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The decrease of the spectral radius, an important characterizer of network dynamics, by removing links is investigated. The minimization of the spectral radius by removing m links is shown to be an NPcomplete problem, which suggests to consider heuristic strategies. Several greedy strategies are compared and several bounds on the decrease of the spectral radius are derived. The strategy that removes that link l = i j with largest product (x1)
On the Randić index of unicyclic graphs
 MATCH Commun. Math. Comput. Chem
"... The Randic ́ index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))− 1 2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertex u and v in G. In the paper, we give sharp lower and upper bounds on the Randic ́ index of unicyclic graphs. ..."
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Cited by 6 (2 self)
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The Randic ́ index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))− 1 2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertex u and v in G. In the paper, we give sharp lower and upper bounds on the Randic ́ index of unicyclic graphs.
The total irregularity of a graph
, 2014
"... In this note a new measure of irregularity of a graph G is introduced. It is named the total irregularity of a graph and is defined as irrt(G) = 12 u,v∈V (G) dG(u) − dG(v)  , where dG(u) denotes the degree of a vertex u ∈ V (G). All graphs with maximal total irregularity are determined. It is als ..."
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Cited by 5 (1 self)
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In this note a new measure of irregularity of a graph G is introduced. It is named the total irregularity of a graph and is defined as irrt(G) = 12 u,v∈V (G) dG(u) − dG(v)  , where dG(u) denotes the degree of a vertex u ∈ V (G). All graphs with maximal total irregularity are determined. It is also shown that among all trees of the same order the star has the maximal total irregularity.
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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Cited by 4 (0 self)
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
Linear inequalities among graph invariants: using GraPHedron to uncover optimal relationships
, 2005
"... Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph nodes, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples ..."
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Cited by 4 (3 self)
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Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph nodes, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalities correspond to the facets of this polytope. They are finite in number, are logically independent, and generate precisely all the linear inequalities valid on the class of graphs. The computer system GraPHedron, developed by some of the authors, is able to produce experimental data about such inequalities for a “small” number of nodes. It greatly helps conjecturing optimal linear inequalities, which are then hopefully proved for any node number. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities in the stability number and the link number are obtained. To this aim, a problem of Ore (1962) related to Turán Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter.
A tight analysis of the maximal matching heuristic
 IN PROC. OF THE ELEVENTH INTERNATIONAL COMPUTING AND COMBINATORICS CONFERENCE (COCOON), LNCS
, 2005
"... We study the algorithm that iteratively removes adjacent vertices from a simple, undirected graph until no edge remains. This algorithm is a wellknown 2approximation to three classical NPhard optimization problems: MINIMUM VERTEX COVER, MINIMUM MAXIMAL MATCHING and MINIMUM EDGE DOMINATING SET. W ..."
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Cited by 3 (2 self)
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We study the algorithm that iteratively removes adjacent vertices from a simple, undirected graph until no edge remains. This algorithm is a wellknown 2approximation to three classical NPhard optimization problems: MINIMUM VERTEX COVER, MINIMUM MAXIMAL MATCHING and MINIMUM EDGE DOMINATING SET. We show that the worstcase 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 ɛdense 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 ɛdense 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.