Nurse Rostering Problem (NRP) or the Nurse Scheduling Problem (NSP) is a complex scheduling problem that affects hospital personnel on a daily basis all over the world and is known to be NP-hard.The problem is to decide which members of a team of nurses should be on duty at any time, during a rostering period of, typically, one month.It is very important to efficiently utilize time and effort, ...

The thesis consists of two parts. In the first part we study two graph transformations,namely the Kelmans transformation and the generalized tree shift. In the second part of thisthesis we study an extremal graph theoretic problem and its relationship with algebraic graphtheory. The main results of this thesis are the following. • We show that the Kelmans transformation is a very ef...

A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a far-reaching generalization of this, conjectured by Wagner: If a class of graphs is minor-closed (i.e., it is...

The “CRC Handbook” is well-known to anyone who has taken a college chemistry course, and CRC Press has traded on this name-familiarity to greatly expand its “Handbook” series. One of the newest entries to join titles such as the Handbook of Combinatorial Designs, the Handbook of Exact Solutions to Ordinary Differential Equations and the Handbook of Edible Weeds, is the Handbook of Graph Theory....

With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices pro...

3 Third Lecture 11 3.1 Applications of the Zarankiewicz Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The Turán Problem for Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 The Girth Problem and Moore’s Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Application of Moore’s Bound to Graph Spanners . . . . . . . . . . . ....

Algebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example, automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects. One of the oldest themes in the area is the investigation of the relation between properties of a graph and the spectrum of its adjacency matrix. A ce...

Understanding the influence of an environment on the evolution of its resident population is a major challenge in evolutionary biology. Great progress has been made in homogeneous population structures while heterogeneous structures have received relatively less attention. Here we present a structured population model where different individuals are best suited to different regions of their env...

A graph G = (V,E) consists of two sets V and E. The elements of V are called the vertices and the elements of E the edges of G. Each edge is a pair of vertices. For instance, the sets V = {1, 2, 3, 4, 5} and E = {{1, 2}, {2, 3}, {3, 4}, {4, 5}} define a graph with 5 vertices and 4 edges. Graphs have natural visual representations in which each vertex is represented by a point and each edge by a...