Identifying codes in vertex-transitive graphs and strongly regular graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs

Asymptotic dimension and asymptotic property

This thesis will be concerned with the study of some ``large-scale'' properties of metric spaces. This area evolved from the study of geometric group theory. Chapter 1 lays out some of the fundamental notions of geometric group theory including information about word metrics, Cayley graphs, quasi-isometries, and ends of groups and graphs. Chapter 2 introduces the idea of ``large-scale'' or ``asymptotic'' properties of metric spaces along the lines proposed by Gromov in cite{Gromov}. After looking at some elementary asymptotic versions of common topological notions, such as connectedness, we focus on asymptotic dimension, the large-scale analog of ordinary covering dimension. In the final chapter, we focus on Dranishnikov's asymptotic version of Haver's property C; see cite{Dranishnikov}. We provide some basic results on metric spaces with asymptotic property C, studying subspaces and unions. We also prove a result involving the product of metric spaces with asymptotic property C and exhibit a metric space with asymptotic property C and infinite asymptotic dimension. In addition, we study the relationships between asymptotic property C and some of our previously introduced concepts such as quasi-isometries and asymptotic dimension

Metric sparsification and operator norm localization

We study an operator norm localization property and its applications to the coarse Novikov conjecture in operator K-theory. A metric space X is said to have operator norm localization property if there exists a positive number c such that for every r>0, there is R>0 for which, if m is a positive locally finite Borel measure on X, H is a separable infinite dimensional Hilbert space and T is a bounded linear operator acting on L^2(X,m) with propagation r, then there exists an unit vector v satisfying with support of diameter at most R and such that |Tv| is larger or equal than c|T|. If X has finite asymptotic dimension, then X has operator norm localization property. In this paper, we introduce a sufficient geometric condition for the operator norm localization property. This is used to give many examples of finitely generated groups with infinite asymptotic dimension and the operator norm localization property. We also show that any sequence of expanding graphs does not possess the operator norm localization property

Identifying codes in vertex-transitive graphs and strongly regular graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V|)+1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order |V|^a with a in {1/4,1/3,2/5}. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs