Random Interval Graphs

In this thesis, which is supervised by Dr. David Penman, we examine random interval graphs. Recall that such a graph is defined by letting $X_{1},\ldots X_{n},Y_{1},\ldots Y_{n}$ be $2n$ independent random variables, with uniform distribution on $[0,1]$. We then say that the $i$th of the $n$ vertices is the interval $[X_{i},Y_{i}]$ if $X_{i}<Y_{i}$ and the interval $[Y_{i},X_{i}]$ if $Y_{i}<X_{i}$. We then say that two vertices are adjacent if and only if the corresponding intervals intersect. We recall from our MA902 essay that fact that in such a graph, each edge arises with probability $2/3$, and use this fact to obtain estimates of the number of edges. Next, we turn to how these edges are spread out, seeing that (for example) the range of degrees for the vertices is much larger than classically, by use of an interesting geometrical lemma. We further investigate the maximum degree, showing it is always very close to the maximum possible value $(n-1)$, and the striking result that it is equal to $(n-1)$ with probability exactly $2/3$. We also recall a result on the minimum degree, and contrast all these results with the much narrower range of values obtained in the alternative \lq comparable\rq\, model $G(n,2/3)$ (defined later). We then study clique numbers, chromatic numbers and independence numbers in the Random Interval Graphs, presenting (for example) a result on independence numbers which is proved by considering the largest chain in the associated interval order. Last, we make some brief remarks about other ways to define random interval graphs, and extensions of random interval graphs, including random dot product graphs and other ways to define random interval graphs. We also discuss some areas these ideas should be usable in. We close with a summary and some comments

Nonautonomous saddle-node bifurcations: random and deterministic forcing

We study the effect of external forcing on the saddle-node bifurcation pattern of interval maps. By replacing fixed points of unperturbed maps by invariant graphs, we obtain direct analogues to the classical result both for random forcing by measure-preserving dynamical systems and for deterministic forcing by homeomorphisms of compact metric spaces. Additional assumptions like ergodicity or minimality of the forcing process then yield further information about the dynamics. The main difference to the unforced situation is that at the critical bifurcation parameter, two alternatives exist. In addition to the possibility of a unique neutral invariant graph, corresponding to a neutral fixed point, a pair of so-called pinched invariant graphs may occur. In quasiperiodically forced systems, these are often referred to as 'strange non-chaotic attractors'. The results on deterministic forcing can be considered as an extension of the work of Novo, Nunez, Obaya and Sanz on nonautonomous convex scalar differential equations. As a by-product, we also give a generalisation of a result by Sturman and Stark on the structure of minimal sets in forced systems.Comment: 17 pages, 5 figure

Modular Decomposition and the Reconstruction Conjecture

We prove that a large family of graphs which are decomposable with respect to the modular decomposition can be reconstructed from their collection of vertex-deleted subgraphs.Comment: 9 pages, 2 figure

Interval-valued fuzzy graphs

We define the Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some of their properties. We also introduce the notion of interval-valued fuzzy complete graphs and present some properties of self complementary and self weak complementary interval-valued fuzzy complete graphs