Cyclic labellings with constraints at two distances

Motivated by problems in radio channel assignment, we consider the vertex-labelling of graphs with non-negative integers. The objective is to minimise the span of the labelling, subject to constraints imposed at graph distances one and two. We show that the minimum span is (up to rounding) a piecewise linear function of the constraints, and give a complete specification, together with associated optimal assignments, for trees and cycles

The Radio Number of Grid Graphs

The radio number problem uses a graph-theoretical model to simulate optimal frequency assignments on wireless networks. A radio labeling of a connected graph $G$ is a function $f:V(G) \to \mathbb Z_{0}^+$ such that for every pair of vertices $u,v \in V(G)$, we have $\lvert f(u)-f(v)\rvert \ge \text{diam}(G) + 1 - d(u,v)$ where $\text{diam}(G)$ denotes the diameter of $G$ and $d(u,v)$ the distance between vertices $u$ and $v$. Let $\text{span}(f)$ be the difference between the greatest label and least label assigned to $V(G)$. Then, the \textit{radio number} of a graph $\text{rn}(G)$ is defined as the minimum value of $\text{span}(f)$ over all radio labelings of $G$. So far, there have been few results on the radio number of the grid graph: In 2009 Calles and Gomez gave an upper and lower bound for square grids, and in 2008 Flores and Lewis were unable to completely determine the radio number of the ladder graph (a 2 by $n$ grid). In this paper, we completely determine the radio number of the grid graph $G_{a,b}$ for $a,b>2$, characterizing three subcases of the problem and providing a closed-form solution to each. These results have implications in the optimization of radio frequency assignment in wireless networks such as cell towers and environmental sensors.Comment: 17 pages, 7 figure

The frequency assignment problem

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight. We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.EPSR

Target formation on the circle by monotone system design

Positivity and Perron-Frobenius theory provide an elegant framework for the convergence analysis of linear consensus algorithms. Here we consider a generalization of these ideas to the analysis of nonlinear consensus algorithms on the circle and establish tools for the design of consensus protocols that monotonically converge to target formations on the circle

Summability of Superstring Theory

Several arguments are given for the summability of the superstring perturbation series. Whereas the Schottky group coordinatization of moduli space may be used to provide refined estimates of large-order bosonic string amplitudes, the super-Schottky group variables define a measure for the supermoduli space integral which leads to upper bounds on superstring scattering amplitudes.Comment: 11 pages, TeX. A remark about C-cycles and dividing cycles and two references have been added to the pape

The frequency assignment problem

This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight. We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.EThOS - Electronic Theses Online ServiceEPSRCGBUnited Kingdo