Graph-theoretic design and analysis of key predistribution schemes

Key predistribution schemes for resource-constrained networks are methods for allocating symmetric keys to de-vices in such a way as to provide an efficient trade-off between key storage, connectivity and resilience. While there have been many suggested constructions for key predistribution schemes, a general understanding of the design prin-ciples on which to base such constructions is somewhat lacking. Indeed even the tools from which to develop such an understanding are currently limited, which results in many relatively ad hoc proposals in the research literature. It has been suggested that a large edge-expansion coefficient in the key graph is desirable for efficient key predistri-bution schemes. However, attempts to create key predistribution schemes from known expander graph constructions have only provided an extreme in the trade-off between connectivity and resilience: namely, they provide perfect resilience at the expense of substantially lower connectivity than can be achieved with the same key storage. Our contribution is two-fold. First, we prove that many existing key predistribution schemes produce key graphs with good expansion. This provides further support and justification for their use, and confirms the validity of expan-sion as a sound design principle. Second, we propose the use of incidence graphs and concurrence graphs as tools to represent, design and analyse key predistribution schemes. We show that these tools can lead to helpful insights and new constructions.

FAULT-TOLERANT METRIC DIMENSION OF CIRCULANT GRAPHS

A set $W$ of vertices in a graph $G$ is called a resolving setfor $G$ if for every pair of distinct vertices $u$ and $v$ of $G$ there exists a vertex $w \in W$ such that the distance between $u$ and $w$ is different from the distance between $v$ and $w$. The cardinality of a minimum resolving set is called the metric dimension of $G$, denoted by $\beta(G)$. A resolving set $W'$ for $G$ is fault-tolerant if $W'\setminus \left\lbrace w\right\rbrace$ for each $w$ in $W'$, is also a resolving set and the fault-tolerant metric dimension of $G$ is the minimum cardinality of such a set, denoted by $\beta'(G)$. The circulant graph is a graph with vertex set $\mathbb{Z}_{n}$, an additive group of integers modulo $n$, and two vertices labeled $i$ and $j$ adjacent if and only if $i -j \left( mod \ n \right)  \in C$, where $C \in \mathbb{Z}_{n}$ has the property that $C=-C$ and $0 \notin C$. The circulant graph is denoted by $X_{n,\bigtriangleup}$ where $\bigtriangleup = \vert C\vert$. In this paper, we study the fault-tolerant metric dimension of a family of circulant graphs $X_{n,3}$ with connection set $C=\lbrace 1,\dfrac{n}{2},n-1\rbrace$ and circulant graphs $X_{n,4}$ with connection set $C=\lbrace \pm 1,\pm 2\rbrace$

Some Problems in Algebraic and Extremal Graph Theory.

In this dissertation, we consider a wide range of problems in algebraic and extremal graph theory. In extremal graph theory, we will prove that the Tree Packing Conjecture is true for all sequences of trees that are \u27almost stars\u27; and we prove that the Erdos-Sos conjecture is true for all graphs G with girth at least 5. We also conjecture that every graph G with minimal degree k and girth at least $2t+1$ contains every tree T of order $kt+1$ such that $\Delta(T)\leq k.$ This conjecture is trivially true for t = 1. We Prove the conjecture is true for t = 2 and that, for this value of t, the conjecture is best possible. We also provide supporting evidence for the conjecture for all other values of t. In algebraic graph theory, we are primarily concerned with isomorphism problems for vertex-transitive graphs, and with calculating automorphism groups of vertex-transitive graphs. We extend Babai\u27s characterization of the Cayley Isomorphism property for Cayley hypergraphs to non-Cayley hypergraphs, and then use this characterization to solve the isomorphism problem for every vertex-transitive graph of order pq, where p and q distinct primes. We also determine the automorphism groups of metacirculant graphs of order pq that are not circulant, allowing us to determine the nonabelian groups of order pq that are Burnside groups. Additionally, we generalize a classical result of Burnside stating that every transitive group G of prime degree p, is doubly transitive or contains a normal Sylow p-subgroup to all p\sp k, provided that the Sylow p-subgroup of G is one of a specified family. We believe that this result is the most significant contained in this dissertation. As a corollary of this result, one easily gives a new proof of Klin and Poschel\u27s result characterizing the automorphism groups of circulant graphs of order p\sp k, where p is an odd prime

Graph-theoretic design and analysis of key predistribution schemes

Key predistribution schemes for resource-constrained networks are methods for allocating symmetric keys to devices in such a way as to provide an efficient trade-off between key storage, connectivity and resilience. While there have been many suggested constructions for key predistribution schemes, a general understanding of the design principles on which to base such constructions is somewhat lacking. Indeed even the tools from which to develop such an understanding are currently limited, which results in many relatively ad hoc proposals in the research literature. It has been suggested that a large edge-expansion coefficient in the key graph is desirable for efficient key predistribution schemes. However, attempts to create key predistribution schemes from known expander graph constructions have only provided an extreme in the trade-off between connectivity and resilience: namely, they provide perfect resilience at the expense of substantially lower connectivity than can be achieved with the same key storage. Our contribution is two-fold. First, we prove that many existing key predistribution schemes produce key graphs with good expansion. This provides further support and justification for their use, and confirms the validity of expansion as a sound design principle. Second, we propose the use of incidence graphs and concurrence graphs as tools to represent, design and analyse key predistribution schemes. We show that these tools can lead to helpful insights and new constructions

The Removal Lemma: algebraic versions and applications

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization