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

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

Metric dimension of Andrásfai graphs

A set $W\subseteq V(G)$ is called a resolving set, if for each pair of distinct vertices $u,v\in V(G)$ there exists $t\in W$ such that $d(u,t)\neq d(v,t)$, where $d(x,y)$ is the distance between vertices $x$ and $y$. The cardinality of a minimum resolving set for $G$ is called the metric dimension of $G$ and is denoted by $\dim_M(G)$. This parameter has many applications in different areas. The problem of finding metric dimension is NP-complete for general graphs but it is determined for trees and some other important families of graphs. In this paper, we determine the exact value of the metric dimension of Andrásfai graphs, their complements and $And(k)\square P_n$. Also, we provide upper and lower bounds for $dim_M(And(k)\square C_n)$

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

In this paper we study the complexity of the following problems: 1. Given a colored graph X=(V,E,c), compute a minimum cardinality set of vertices S (subset of V) such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G <= S_n given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT~algorithms. 2. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c)compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time, while starting from color class size 4 they become W[P]-hard

A Study of the Relationship Between Spectrum and Geometry Through Fourier Frames and Laplacian Eigenmaps

This thesis has two parts. The first part is a study of Fourier frames. We follow the development of the theory, beginning with its classical roots in non-uniform sampling in Paley-Wiener Spaces, to its current state, the study of the spectral properties of finite measures on locally compact abelian groups. The aim of our study is to understand the relationship between the geometry of the supporting set of a measure and the spectral properties it exhibits. In the second part, we study extensions of the Laplacian Eigenmaps algorithm and their uses in hyperspectral image analysis. In particular, we show that there is a natural way of including spatial information in the analysis that improves classification results. We also provide evidence supporting the use of Schrödinger Eigenmaps as a semisupervised tool for feature extraction. Finally, we show that Schrödinger Eigenmaps provides a platform for fusing Laplacian Eigenmaps with other clustering techniques, such as kmeans clustering