We consider uniform random walks on finite graphs with n nodes. When the hitting times are symmetric, the expected covering time is at least 89 log n O(n log log n) uniformly over all such graphs. We also obtain bounds for the covering times in terms of the eigenvalues of the transition matrix of the Markov chain. For distance-regular graphs, a general lower bound of ( n 1 ) l o g n is obtained...

In this article, we introduce the concept of co-covering which is the dual of covering concept. Then, we prove several theorems being similar to the theorems that have been de-veloped for the covering concept. For example, we provide the lifting criterion for co-coverings of a topological space X, which helps us to classify them by subgroups of the group of all homeomorphisms of X.

We prove that maximum three dimensional matching is a MAX SNP-hard problem. If the number of occurrences of elements in triples is bounded by a constant the problem is MAX SNP-complete. As corollaries we prove that maximum covering by 3-sets and maximum covering of a graph by triangles are MAX SNP-hard. The problems are MAX SNP-complete if the number of occurrences of the elements and the degre...

The Delay Constrained Relay Node Placement (DCRNP) problem in Wireless Sensor Networks (WSNs) aims to deploy minimum relay nodes such that for each sensor node there is a path connecting this sensor node to the sink without violating delay constraint. As WSNs are gradually employed in time-critical applications, the importance of the DCRNP problem becomes noticeable. For the NP-hard nature of D...

Multi-hop ad-hoc networks consist of nodes which cooperate by forwarding packets for each other to allow communication beyond the power range of each node. In pure ad-hoc networks, no additional infrastructure is required to allow the nodes to communicate. Multi-hop hybrid networks are a combination of ad-hoc and cellular networks. As in ad-hoc networks, the nodes forward packets on behalf of o...

This paper concerns a class of maximum covering location problems in networks in uncertain environments. It is assumed that: (a) relative weights of demand nodes are either deterministic or imprecise, described by linguistic expressions, (b) potential facility site locations are limited to network nodes. The concept of coverage is extended to include a degree of node coverage which means that t...

Covering Tour Problem (CTP) is the generalized form of Traveling Salesman Problem (TSP), which has found different applications in the designing of distribution networks, disaster relief, and transportation routing. The purpose of this problem is to determine the Hamiltoniancyclewiththe lowest costusinga subset of all the nodes, such that the other nodes would be in a distance shorter than the ...

In this survey, we provide the most important computational methods to find the PageRank. This is a new comprehensive review of all major issues which are associated with PageRank problem, covering the basic topics, the iterative methods, lumping of nodes, the modification of lumping the nodes, rank-one perturbation, rank-r perturbation, advanced numerical linear algebra methods, conditioning, ...

In a classical covering problem, we are given a set of requests that we need to satisfy (fully or partially), by buying a subset of items at minimum cost. For example, in the k-MST problem we want to find the cheapest tree spanning at least k nodes of an edge-weighted graph. Here nodes and edges represent requests and items, respectively. In this paper, we initiate the study of a new family of ...

For a finite connected graph let be the spectral radius of its universal cover. We prove that for any graph of average degree and derive from it the following generalization of the Alon Boppana bound. If the average degree of the graph after deleting any radius ball is at least , then its second largest eigenvalue in absolute value is at least for some absolute constant . This result is tight i...