work. The method of modeling and the contents of our articles [SL1], [SL2], [RIS1], [PB3], and [PB4] will be explained below. 2. The Method of Modeling

In this research, graph theory is applied for two purposes. One is to express the routing of communication and the other is to express the condition of roads. In both case, we require a directed graph in which each graph edge is replaced by a directed graph edge. Then to be able to add a value to each route, a weighted graph in which each graph edge has an associated numerical value is adopted in this research as well. In a communication route graph, the weight of an edge represents the intensity of an electric wave, the condition of a battery, and the moving speed of each mobile terminal. In road graphs, the weight of an edge represents bend conditions of a road, distances between crossings, and conditional probabilities of advancing directions based on the road situation. Moreover, to be able to improve the efficiency of routing, each mobile terminal needs to have information about mobile terminals of the neighborhood which can communicate directly with it. For this reason, we also add neighborhood information to each vertex in the graph of communication routing. The neighborhood of a vertex v in a graph is the set of all the vertices adjacent to v. In the example of Fig. 1, when terminal � wants to communicate with terminal � outside the area in which it can communicate directly, it is necessary to look for a terminal which can relay the communication. A linear search algorithm tracing each edge going out from � is the simplest way to find such a terminal, but it has an inefficient calculation cost. As opposed to this it is more efficient to find routing paths by using the information of vertex neighborhoods. In this example, by merely taking the union operation of the neighborhood of � and �, it becomes immediately clear that � is a relay terminal between � and �

Minimum energy disjoint path routing in wireless ad-hoc networks

We develop algorithms for finding minimum energy disjoint paths in an all-wireless network, for both the node and linkdisjoint cases. Our major results include a novel polynomial time algorithm that optimally solves the minimum energy 2 link-disjoint paths problem, as well as a polynomial time algorithm for the minimum energy k node-disjoint paths problem. In addition, we present efficient heuristic algorithms for both problems. Our results show that link-disjoint paths consume substantially less energy than node-disjoint paths. We also found that the incremental energy of additional linkdisjoint paths is decreasing. This finding is somewhat surprising due to the fact that in general networks additional paths are typically longer than the shortest path. However, in a wireless network, additional paths can be obtained at lower energy due to the broadcast nature of the wireless medium. Finally, we discuss issues regarding distributed implementation and present distributed versions of the optimal centralized algorithms presented in the paper

CHAPTERS

CHAPTERS 1. GRAPH THEORY CONCEPTS 1.1. GRAPHS 1.2. PROPERTIES OF GRAPHS 2. SEMIREGULAR GRAPHS 2.1

Graph Theory. 2. Vertex Descriptors and Graph Coloring

p. 37-52 This original work presents the construction of a set of ten sequence matrices and their applications for ordering vertices in graphs. For every sequence matrix three ordering criteria are applied: lexicographic ordering, based on strings of numbers, corresponding to every vertex, extracted as rows from sequence matrices; ordering by the sum of path lengths from a given vertex; and ordering by the sum of paths, starting from a given vertex. We also examine a graph that has different orderings for the above criteria. We then proceed to demonstrate that every criterion induced its own partition of graph vertex. We propose the following theoretical result: both LAVS and LVDS criteria generate identical partitioning of vertices in any graph. Finally, a coloring of graph vertices according to introduced ordering criteria was proposed