Aleksandar Ilić

A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem

We present a new general variable neighborhood search approach for the uncapacitated single allocation p-hub median problem in networks. This NP hard problem is concerned with locating hub facilities in order to minimize the traffic between all origin-destination pairs. We use three neighborhoods and efficiently update data structures for calculating new total flow in the network. In addition to the usual sequential strategy, a new nested strategy is proposed in designing a deterministic variable neighborhood descent local search. Our experimentation shows that general variable neighborhood search based heuristics outperform the best-known heuristics in terms of solution quality and computational effort. Moreover, we improve the best-known objective values for some large Australia Post and PlanetLab instances. Results with the new nested variable neighborhood descent show the best performance in solving very large test instances.

A general variable neighborhood search for the one-commodity pickup-and-delivery travelling salesman problem

We present a variable neighborhood search approach for solving the one-commodity pickup-and-delivery travelling salesman problem. It is characterized by a set of customers such that each of the customers either supplies (pickup customers) or demands (delivery customers) a given amount of a single product, and by a vehicle, whose given capacity must not be exceeded, that starts at the depot and must visit each customer only once. The objective is to minimize the total length of the tour. Thus, the considered problem includes checking the existence of a feasible travelling salesman’s tour and designing the optimal travelling salesman’s tour, which are both NP-hard problems. We adapt a collection of neighborhood structures, k-opt, double-bridge and insertion operators mainly used for solving the classical travelling salesman problem. A binary indexed tree data structure is used, which enables efficient feasibility checking and updating of solutions in these neighborhoods. Our extensive computational analysis shows that the proposed variable neighborhood search based heuristics outperforms the best-known algorithms in terms of both the solution quality and computational efforts. Moreover, we improve the best-known solutions of all benchmark instances from the literature (with 200 to 500 customers). We are also able to solve instances with up to 1000 customers.

Distance spectral radius of trees with fixed maximum degree

C ‡ Abstract. Distance energy is a newly introduced molecular graph-based analog of the total �-electron energy, and it is defined as the sum of the absolute eigenvalues of the molecular distance matrix. For trees and unicyclic graphs, distance energy is equal to the doubled value of the distance spectral radius. In this paper, we introduce a general transformation that increases the distance spectral radius and provide an alternative proof that the path Pn has the maximal distance spectral radius among trees on n vertices. Among the trees with a fixed maximum degree �, we prove that the broom Bn,� (consisting of a star S�+1 and a path of length n � 1 attached to an arbitrary pendent vertex of the star) is the unique tree that maximizes the distance spectral radius, and conjecture the structure of a tree which minimizes the distance spectral radius. As a first step towards this conjecture, we characterize the starlike trees with the minimum distance spectral radius.

Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number☆

Abstract In this paper, we present sharp bounds for the Zagreb indices, Harary index and hyper-Wiener index of graphs with a given matching number, and we also completely determine the extremal graphs.

Generalized Fibonacci cubes

Generalized Fibonacci cube Q d ( f ) is introduced as the graph obtained from the d -cube Q d by removing all vertices that contain a given binary string f as a substring. In this notation, the Fibonacci cube ? d is Q d ( 11 ) . The question whether Q d ( f ) is an isometric subgraph of Q d is studied. Embeddable and non-embeddable infinite series are given. The question is completely solved for strings f of length at most five and for strings consisting of at most three blocks. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial cases Q d ( 10 ) and Q d ( 01 ) , the only generalized Fibonacci cubes that are median closed subgraphs of the corresponding hypercubes. For admissible strings f , the f -dimension of a graph is introduced. Several problems and conjectures are also listed.

On the clique number of integral circulant graphs

Abstract The concept of gcd-graphs is introduced by Klotz and Sander; they arise as a generalization of unitary Cayley graphs. The gcd-graph X n ( d 1 , … , d k ) has vertices 0 , 1 , … , n − 1 , and two vertices x and y are adjacent iff gcd ( x − y , n ) ∈ D = { d 1 , d 2 , … , d k } . These graphs are exactly the same as circulant graphs with integral eigenvalues characterized by So. In this work we deal with the clique number of integral circulant graphs and investigate the conjecture proposed in [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, The Electronic Journal of Combinatorics 14 (2007) #R45] that the clique number divides the number of vertices in the graph X n ( D ) . We completely solve the problem of finding the clique number for integral circulant graphs with exactly one and two divisors. For k ⩾ 3 , we construct a family of counterexamples and disprove the conjecture in this case.

On distance-balanced graphs

It is shown that the graphs for which the Szeged index equals @?G@?@?|G|^24 are precisely connected, bipartite, distance-balanced graphs. This enables us to disprove a conjecture proposed in [M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149-1163]. Infinite families of counterexamples are based on the Handa graph, the Folkman graph, and the Cartesian product of graph. Infinite families of distance-balanced, non-regular graphs that are prime with respect to the Cartesian product are also constructed.

Generalizations of Wiener Polarity Index and Terminal Wiener Index

In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. We introduce a generalized Wiener polarity index Wk (G) as the number of unordered pairs of vertices {u, v} of G such that the shortest distance d (u, v) between u and v is k (this is actually the kth coefficient in the Wiener polynomial). For k = 3, we get standard Wiener polarity index. Furthermore, we generalize the terminal Wiener index TWk (G) as the sum of distances between all pairs of vertices of degree k. For k = 1, we get standard terminal Wiener index. In this paper we describe a linear time algorithm for computing these indices for trees and partial cubes, and characterize extremal trees maximizing the generalized Wiener polarity index and generalized terminal Wiener index among all trees of given order n.

On the extremal properties of the average eccentricity

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc(G) of a graph G is the mean value of eccentricities of all vertices of G. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease ecc(G). Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (the clique number and the independence number), refute one AutoGraphiX conjecture about the average eccentricity and the minimum vertex degree and correct one AutoGraphiX conjecture about the domination number.

Network analysis using a novel highly discriminating topological index

When characterizing networks structurally, the discriminating ability of a topological measure (or also called index) is crucial. This relates to investigate its discrimination power (or also called uniqueness or degeneracy) that indicates how meaningful the given measure can distinguish nonisomorphic networks. In terms of biological and chemical graph analysis, a highly discriminative measure is desirable because it then has the ability to detect minor structural changes within the given network. In this article, the discriminating ability of a new super index based on Shell-matrices and polynomials is tested using (real) atomic and synthetic structures. As a result, the new descriptor can distinguish all graphs uniquely. We emphasize that some molecular descriptors which are embedded in the super index have already shown excellent correlating ability with alkanes properties. In view of the fact that most of the existing topological graph measures are degenerated, the new super index seems to be a good starting point for performing further studies in the context of network analysis. In the future, we also intent to use other sets of networks, for instance molecular graphs, to further examine the index and its meaning. In particular, we emphasize that only those indices which possess low computational complexity do have the potential to be applied for analyzing complex systems properly.