Mathematics

Network computer connections are growing at a faster rate. There is a needs advanced algorithms that helps in optimizing the network performance. Classical optimization based approaches have been applied since long time. But there is a need for more flexible and advanced algorithmic approaches to model the increasingly complex networks. Optimization is to find the best solution, either maximum or minimum out of all feasible solutions. In this paper, we propose an extension Travelling salesman Problem (TSP) in a systemic way by means of the modified version of the Multiple Traveling Salesman Problem (MTSP). The aim of this paper is used to solve the routing problem by considering two objectives; one is the minimization of the total time travelled by the salesman considering the capacity of the edges. After providing the offline version of the problem we propose and prove the online version of the problem using the Dynamic capacity Routing (DCR) algorithm. This work gives all feasible tools that can help to solve the problem and provide experimental insights into feasible network instances based on real life and simulated data.

Connected Dominating Sets (CDSs) can serve as virtual backbones for wireless networks. A smaller virtual backbone (minimum size CDS) incurs less communication overhead. Unfortunately, computing a minimum size CDS is NP-hard and thus many algorithms were developed which concentrates on how to construct smaller CDSs. Aiming for minimum number of nodes in CDS, results in CDSs that are highly unstable. Here we present four CDS algorithms which are constructed based on strategy, density, Threshold Neighborhood Distance Ratio (TNDR) and velocity. Any node in the network can be a strategic node depending upon the application and need and will be selected as the starting node instead of the node with highest density which concentrates on minimum size CDS. Simulation methods are used to establish the efficiency of the proposed algorithm.

This paper deals with a new interconnection network motivated by molecular structure of a chemical compound SiO 4. The different forms of silicate available in nature lead to the introduction of the (DSL) dominating silicate network. The first section deals with the introduction to (Resolving number) minimum metric dimension problems, and few related work about Silicate networks. The second section introduces and gives an account of the proof to the topological properties of poly-oxide, poly-silicate, dominating oxide (DOX), dominating silicate networks, and regular triangulene oxide network (RTOX). The third section deals with the drawing algorithm for dominating silicate network, and shown complete embedding of oxide, silicate network in to dominating oxide, dominating Silicate network respectively. The fourth section contains the proof of the minimum metric dimension of regular triangulene oxide network to be 2.

A minimum resolving set or a metric basis M for a graph G(V, E) is a small subset of vertices of V such that for every pair of vertices x and y of V \ M, there exist at least one vertex m in M such that the distance between x and m is not equal to the distance between y and m. The number of elements of the metric basis M of G is called metric dimension and the elements of a metric basis are called landmarks. A metric dimension problem for a graph G is to find a metric basis for G. In this paper a new silicate graph called Silicate Stars or Star of Silicate Networks SSL(n) has been derived from Star of David Networks SD(n). The metric dimension problem has been solved for SSL(n) , Single Oxide chain, and single Silicate chain. The problem of finding the metric dimension of a general graph is an NP Complete Problem.

In this paper, we have introduced few Interconnection Networks, called David Derived Network DD(n) , Dominating David Derived Network DDD(n), Honeycomb cup Network HCC(n) and Kite Regular Trianguline Mesh KRrTM(n). We have given drawing algorithm for DDD(n) from Honeycomb network HC(n) and embedded poly-Honeycomb Networks, KRrTM(n) in to Dominating David Derived Networks. Also we have investigated the metric dimension of Star of David network SD(n) and lower bound of the metric dimension for DD(n).

The Multiple Travelling Salesman Problems (MTSP) can be used in a wide range of discrete optimization problems. As the solution to this problem has wide applicability in many practical fields, this NP Hard problem highly raises the need for an efficient solution. The problem is determining a set of routes for the salesmen that jointly visit a set of given cities which are facing difficulty because of road congestion. Selection of proper route is based on the road capacity, which is the deciding factor in the opt vehicle usage. The objective of the study is to optimize the vehicle utilization and minimize the time of travel by salesman based on the road capacity. The solution to this problem is achieved in 3 steps; the first step is by assigning addresses to cities by Ad-assignment algorithm. The second step is by assigning cities and vehicles to salesman by Sl-assignment algorithm. The third step is by using Parallel Shortest Path Multiple Salesman (PSPMS) algorithms to obtain the shortest path. The PSPMS algorithm runs in parallel for each salesman. The solutions to the problem are known to possess an exponential time complexity. From the result we observe that PSPMS is one of the best approximate algorithms used to solve MTSP.

This paper deals with the survey of basic networks like Spidergon network and Honeycomb Torus network with respect to network cost. The basic network can be modelled in different structures to overcome the problems in handling the user density patterns, utilizing the available bandwidth effectively and improvising the efficiency. These bottlenecks are overcome by selecting the proper structure for a particular network based on network cost. In this paper the efficiency of specified network is estimated with respect to the number of nodes.

Efficient routing between nodes is the most important challenge in a Mobile Ad Hoc Network (MANET). A Connected Dominating Set (CDS) acts as a virtual backbone for routing in a MANET. Hence, the construction of CDS based on the need and its application play a vital role in the applications of the MANET. The PipeLined Strategic CDS (PLS-CDS) is constructed based on strategy, dynamic diameter and transmission range. The strategy used for selecting the starting node is any source node in the network, which has its entire destination within a virtual pipelined coverage, instead of the node with maximum connectivity. The other nodes are then selected based on density and velocity. The proposed CDS also utilizes the energy of the nodes in the network in an optimized manner. Simulation results showed that the proposed algorithm is better in terms of size of the CDS and average hop per path length.