Dans la nature existent de nombreux exemples de systèmes dynamiques complexes: systèmes neuronaux, communautés, écosystèmes, réseaux de régulation génétiques, etc. Ces derniers, en particulier, sont de notre intérêt et sont souvent modélisés par des réseaux booléens. Un réseau booléenne peut être considérée comme un digraphe, où les sommets correspondent à des gènes ou de produits de gènes, tandis que les arcs indiquent les interactions entre eux. Une niveau d'expression des gènes est modélisé par des valeurs binaires, 0 ou 1, indiquant deux états de la transcription, soit activité, soit inactivité, respectivement, et ce niveau change dans le temps selon certains fonction locaux d'activation qui dépend des états d'un ensemble de nœuds (les gènes). L'effet conjoint des fonctions d'activation locale définit une fonction de transition globale: ainsi, le autre élément nécessaire dans la description du modèle est fonction de mise à jour, qui détermine quand chaque nœud doit être mis à jour, et donc, comme les fonctions local se combinent dans une fonction globale (en d'autres termes, il doit décrire les temps relative de les activités régulatoires). Comme un réseau booléen avec n sommets a 2 ^ n états globaux, à partir d'un état ​​de départ, et dans un nombre fini de mises à jour, le réseau atteindra un fixe point ou un cycle limite, appelée attracteurs qui sont souvent associées à des phénotypes distincts (états-cellulaire) définis par les patrons d'activité des gènes. Un réseau de régulation Booléenne (REBN) est un réseau Booléen où chaque interaction entre les éléments de la réseau correspond soit à une interaction positif ou d'une interaction négative. Ainsi, le digraphe interaction associée à une REBN est un digraphe signé où un circuit est appelé positif (négatif) si le nombre de ses arcs négative est pair (impair). Dans ce contexte, il y a diverses études sur l'importance du les circuits positif et négatifs dans le comportement dynamique de différents systèmes en Biologie. En effet le point de départ de cette thèse est basée sur un résultat en disant que le nombre maximal de points fixes d'une REBN dépend d'un ensemble de cardinalité minimale qu'intersecte tous les cycles positifs (également dénommés positive feedback vertex set) du digraphe signé associé. D'autre part, un autre aspect important de circuits est leur rôle dans la robustesse des réseaux booléens par rapport différents types de mise à jour déterministe. Dans ce contexte, un élément clé mathématique est le update digraphe qui est un digraphe étiqueté associé à la réseau dont les étiquettes sur les arcs sont définies comme suit: un arc (u,v) est dit être positif si l'état de sommet u est mis à jour en même temps ou après que celle de v, et négative sinon. Ainsi, un cycle dans le digraphe étiqueté est dite positive (négative) si tous ses arcs sont positifs (négatifs). Cela laisse en évidence que parler de "positif" et "négatif" a des significations différentes selon le contex: digraphes signé ou digraphes étiquetés. Ainsi, nous allons voir dans cette thèse, les relations entre les feedback sets et la dynamique des réseaux Booléens à travers l'étude analytique de ces deux fondamentaux objets mathématiques: le digraphe (de connexion) signé et l'update digraphe. / In the nature there exist numerous examples of complex dynamical systems: neural systems, communities, ecosystems, genetic regulatory networks, etc. These latest, in particular are of our interest and are often modeled by Boolean networks. A Boolean network can be viewed as a digraph, where the vertices correspond to genes or gene products, while the arcs denote interactions among them. A gene expression level is modeled by binary values, 0 or 1, indicating two transcriptional states, either active or inactive, respectively, and this level changes in time according to some local activation function which depends on the states of a set of nodes (genes). The joint effect of the local activation functions defines a global transition function; thus, the other element required in the description of the model is an update schedule which determines when each node has to be updated, and hence, how the local functions combine into the global one (in other words, it must describe the relative timings of the regulatory activities). Since a Boolean network with n vertices has 2^n global states, from a starting state, within a finite number of udpates, the network will reach a fixed point or a limit cycle, called attractors that are often associated to distinct phenotypes (cellular states) defined by patterns of gene activity. A regulatory Boolean network (REBN) is a Boolean network where each interaction between the elements of the network corresponds either to a positive or to a negative interaction. Thus, the interaction digraph associated to a REBN is a signed digraph where a circuit is called positive (negative) if the number of its negative arcs is even (odd). In this context, there are diverse studies about the importance of the positive and negative circuits in the dynamical behavior of different systems in Biology. Indeed the starting point of this Thesis is based on a result saying that the maximum number of fixed points of a REBN depends on a minimum cardinality vertex set whose elements intersects to all the positive cycles (also named a positive feedback vertex set) of the associated signed digraph. On the other hand, another important aspect of circuits is their role in the robustness of Boolean networks with respect to different deterministic update schedules. In this context a key mathematical element is the update digraph which is a labeled digraph associated to the network and whose labels on the arcs are defined as follows: an arc (u,v) is said to be positive if the state of vertex u is updated at the same time or after than that of v, and negative otherwise. Hence, a cycle in the labeled digraph is called positive (negative) if all its arcs are positive (negative). This leaves in evidence that talk of "positive" and "negative" has different meanings depending on the contex: signed digraphs or labeled digraphs. Thus, we will see in this thesis, relationships between feedback sets and the dynamics of Boolean networks through the analytical study of these two fundamental mathematical objects: the signed (connection) digraph and the update digraph.

This thesis generalizes to digraphs certain recent results about graphs. There are special digraphs C such that AxC is isomorphic to BxC for some pair of distinct digraphs A and B. Lovasz named these digraphs C zero-divisors and completely characterized their structure. Knowing that all directed cycles are zero-divisors, we focus on the following problem: Given any directed cycle D and any digraph A, enumerate all digraphs B such that AxD is isomorphic to BxD. From our result for cycles, we generalize to an arbitrary zero-divisor C, developing upper and lower bounds for the collection of digraphs B satisfying AxC isomorphic to BxC.

There are four prominent product graphs in graph theory: Cartesian, strong, direct, and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted $G \circ H$, for some digraphs $G$ and $H$. The paper concludes by proving a cancellation property for the lexicographic product of digraphs $G$, $H$, $A$, and $B$: if $G \circ H \cong A \circ B$ and $|V(G)| = |V(A)|$, then $G \cong A$. It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments.

A cancellation graph G is one for which given any graph C, we have G\times C\cong X\times C implies G\cong X. In this thesis, we characterize all bipartite cancellation graphs. In addition, we characterize all solutions X to G\times C\cong X\times C for bipartite G. A characterization of non-bipartite cancellation graphs is yet to be found. We present some examples of solutions X to G\times C\cong X\times C for non-bipartite G, an example of a non-bipartite cancellation graph, and a conjecture regarding non-bipartite cancellation graphs.

This thesis investigates the average consensus of multi-agent systems with linear dynamics whose interconnections are modelled by balanced digraphs with matrix- weights. The thesis first introduces the notion of balanced digraphs and mirror graphs for matrix weights. Then it proves that the matrix-weight-balanced con- sensus controller is indeed globally asymptotically stable. The Lyapunov stability analysis exploits the properties of the mirror graph of a balanced digraph. Further, the necessary and sufficient condition for the system to achieve average consensus is shown to be positive definiteness of the matrix weights of its balanced digraph. Simulations with robots in SIMULINK verify that positive definite matrix weights on balanced graphs are indeed necessary and sufficient for average consensus. Fi- nally formation control of a multi-robot system is shown to be an application of the matrix-weight-balanced consensus algorithm using real time simulation of Clearpath Ridgeback robots in Gazebo and ROS. / Graduate

Graphs provide interesting ways to generate families of lattice polytopes. In particular, one can use matrices encoding the information of a finite graph to define vertices of a polytope. This dissertation initiates the study of the Laplacian simplex, PG, obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. The Laplacian simplex is extended through the use of a parallel construction with a finite digraph D to obtain the Laplacian polytope, PD. Basic properties of both families of simplices, PG and PD, are established using techniques from Ehrhart theory. Motivated by a well-known conjecture in the field, our investigation focuses on reflexivity, the integer decomposition property, and unimodality of Ehrhart h*-vectors of these polytopes. A systematic investigation of PG for trees, cycles, and complete graphs is provided, which is enhanced by an investigation of PD for cyclic digraphs. We form intriguing connections with other families of simplices and produce G and D such that the h*-vectors of PG and PD exhibit extremal behavior.

This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any operator algebra A, the equality A ⊗_max B = A ⊗_min B either implies, or is implied by, the equalities A ⊗_max C_i = A ⊗_min C_i for every i. These results can be applied to the setting of a discrete group G pre-ordered by a subsemigroup G⁺, where B ⊆ C*_r(G) is the subalgebra of the reduced group C*-algebra of G generated by G⁺, and C_i = A(Q_i) are digraph algebras defined by considering certain pre-ordered subsets Q_i of G. The 16-dimensional algebra A₄ of real sedenions is obtained by applying the Cayley-Dickson doubling process to the real division algebra of octonions. The classification of subalgebras of A₄ up to conjugacy (i.e. by the action of the automorphism group of A₄) was completed in a previous investigation, except for the collection of those subalgebras which are isomorphic to the quaternions. We present a classification of quaternion subalgebras up to conjugacy.

Slices are digraphs that can be composed together to form larger digraphs.In this thesis we introduce the foundations of a theory whose aim is to provide ways of defining and manipulating infinite families of combinatorial objects such as graphs, partial orders, logical equations etc. We give special attentionto objects that can be represented as sequences of slices. We have successfully applied our theory to obtain novel results in three fields: concurrency theory,combinatorics and logic. Some notable results are: Concurrency Theory: We prove that inclusion and emptiness of intersection of the causalbehavior of bounded Petri nets are decidable. These problems had been open for almost two decades. We introduce an algorithm to transitively reduce infinite familiesof DAGs. This algorithm allows us to operate with partial order languages defined via distinct formalisms, such as, Mazurkiewicztrace languages and message sequence chart languages. Combinatorics: For each constant z ∈ N, we define the notion of z-topological or-der for digraphs, and use it as a point of connection between the monadic second order logic of graphs and directed width measures, such as directed path-width and cycle-rank. Through this connection we establish the polynomial time solvability of a large numberof natural counting problems on digraphs admitting z-topological orderings. Logic: We introduce an ordered version of equational logic. We show thatthe validity problem for this logic is fixed parameter tractable withrespect to the depth of the proof DAG, and solvable in polynomial time with respect to several notions of width of the equations being proved. In this way we establish the polynomial time provability of equations that can be out of reach of techniques based on completion and heuristic search. / <p>QC 20131120</p>

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In this thesis we examine the computational complexity of certain digraph homomorphism problems. A homomorphism between digraphs, denoted by $f: G \to H$, is a mapping from the vertices of $G$ to the vertices of $H$ such that the arcs of $G$ are preserved. The problem of deciding whether a homomorphism to a fixed digraph $H$ exists is known as the $H$-colouring problem. We prove a generalization of a theorem due to Bang-Jensen, Hell and MacGillivray. Their theorem shows that for every semi-complete digraph $H$, $H$-colouring exhibits a dichotomy: $H$-colouring is either polynomial time solvable or it is NP-complete. We show that the class of local tournaments also exhibit a dichotomy. The NP-completeness results are found using direct NP-completeness reductions, indicator and vertex (and arc) sub-indicator constructions. The polynomial cases are handled by appealing to a result of Gutjhar, Woeginger and Welzl: the \underbar{$X$}-graft extension. We also provide a new proof of their result that follows directly from the consistency check. An unexpected result is the existence of unicyclic local tournaments with NP-complete homomorphism problems. During the last decade a new approach to studying the complexity of digraph homomorphism problems has emerged. This approach focuses attention on so-called polymorphisms as a measure of the complexity of a digraph homomorphism problem. For a digraph $H$, a polymorphism of arity $k$ is a homomorphism $f: H^k \to H$. Certain special polymorphisms are conjectured to be the key to understanding $H$-colouring problems. These polymorphisms are known as weak near unanimity functions (WNUFs). A WNUF of arity $k$ is a polymorphism $f: H^k \to H$ such that $f$ is idempotent an $f(y,x,x,\ldots,x)=f(x,y,x,\ldots,x)=f(x,x,y,\ldots,x) = \cdots = f(x,x,x,\ldots,y)$. We prove that a large class of polynomial time $H$-colouring problems all have a $\WNUF$. Furthermore we also prove some non-existence results for $\WNUF$s on certain digraphs. In proving these results, we develop a vertex (and arc) sub-indicator construction as well as an indicator construction in analogy with the ones developed by Hell and Ne{\v{s}}et{\v{r}}il. This is then used to show that all tournaments with at least two cycles do not admit a $\WNUF_k$ for $k>1$. This furnishes a new proof (in the case of tournaments) of the result by Bang-Jensen, Hell and MacGillivray referred to at the start. These results lend some support to the conjecture that $\WNUF$s are the ``right'' functions for measuring the complexity of $H$-colouring problems. We also study a related notion, namely that of an injective homomorphism. A homomorphism $f: G \to H$ is injective if the restriction of $f$ to the in-neighbours of every vertex in $G$ is an injective mapping. In order to classify the complexity of these problems we develop an indicator construction that is suited to injective homomorphism problems. For this type of digraph homomorphism problem we consider two cases: reflexive and irreflexive targets. In the case of reflexive targets we are able to classify all injective homomorphism problems as either belonging to the class of polynomial time solvable problems or as being NP-complete. Irreflexive targets pose more of a problem. The problem lies with targets of maximum in-degree equal to two. Targets with maximum in-degree one are polynomial, while targets with in-degree at least three are NP-complete. There is a transformation from (ordinary) graph homomorphism problems to injective, in-degree two, homomorphism problems (a reverse transformation also exists). This transformation provides some explanation as to the difficulty of the in-degree two case. We nonetheless classify all injective homomorphisms to irreflexive tournaments as either being a problem in P or a problem in the class of NP-complete problems. We also discuss some upper bounds on the injective oriented irreflexive (reflexive) chromatic number.