Let $M_v$ denotes a complete mixed graph on $v$ vertices, and let $S_6^i$ denotes the partial orientation of the 6-star with twice as many arcs as edges. In this work, we state and prove the necessary and sufficient conditions for the existence of $\lambda$-fold decomposition of a complete mixed graph into $S_6^i$ for $i\in\{1,2,3,4\}$. We used the difference method for our proof in some cases. We also give some general sufficient conditions for the existence of $S_6^i$-decomposition of the complete bipartite mixed graph for $i\in\{1,2,3,4\}$. Finally, this work introduces the decomposition of a complete mixed graph with a hole into mixed stars.

Criminal networks exploit vulnerabilities in the global financial system, using it as a conduit to launder criminal proceeds. Law enforcement agencies, financial institutions, and regulatory organizations often scrutinize voluminous financial records for suspicious activities and criminal conduct as part of anti-money laundering investigations. However, such studies are narrowly focused on incidents and triggered by tip-offs rather than data mining insights. This research models cross-border financial flows using a network theoretic approach and proposes a symmetric-key encryption algorithm to preserve information privacy in multi-dimensional data sets. The newly developed tools will enable regulatory organizations, financial institutions, and law enforcement agencies to identify suspicious activity and criminal conduct in cross-border financial transactions. Anti-money laundering, which comprises laws, regulations, and procedures to combat money laundering, requires financial institutions to verify and identify their customers in various circumstances and monitor suspicious activity transactions. Instituting anti-money laundering laws and regulations in a country carries the benefit of creating a data-rich environment, thereby facilitating non-classical analytical strategies and tools. Graph theory offers an elegant way of representing cross-border payments/receipts between resident and non-resident parties (nodes), with links representing the parties' transactions. The network representations provide potent data mining tools, facilitating a better understanding of transactional patterns that may constitute suspicious transactions and criminal conduct. Using network science to analyze large and complex data sets to detect anomalies in the data set is fast becoming more important and exciting than merely learning about its structure. This research leverages advanced technology to construct and visualize the cross-border financial flows' network structure, using a directed and dual-weighted bipartite graph. Furthermore, the develops a centrality measure for the proposed cross-border financial flows network using a method based on matrix multiplication to answer the question, "Which resident/non-resident nodes are the most important in the cross-border financial flows network?" The answer to this question provides data mining insights about the network structure. The proposed network structure, centrality measure, and characterization using degree distributions can enable financial institutions and regulatory organizations to identify dominant nodes in complex multi-dimensional data sets. Most importantly, the results showed that the research provides transaction monitoring capabilities that allow the setting of customer segmentation criteria, complementing the built-in transaction-specific triggers methods for detecting suspicious activity transactions. / Thesis (PhD)--University of Pretoria, 2021. / Banking Sector Education and Training Authority (BANKSETA) / UP Postgraduate Bursary / Industrial and Systems Engineering / PhD / Unrestricted

Necessary and sufficient conditions are given for the existence of isomorphic decompositions of the complete bipartite graph, the complete graph with a hole, and the λ-fold complete graph into copies of a 4-cycle with a pendant edge.

Dans cette thèse, nous définissons une nouvelle classe de graphes : les graphes hypotriangulés. Les graphes hypotriangulés vérifient que pour tout chemin de longueur deux, il existe une arête ou un autre chemin de longueur deux entre ses extrémités. Cette classe permet par exemple de modéliser des réseaux robustes. En effet, nous montrons que dans de tels graphes, la suppression d'une arête ou d'un sommet ne modifie pas la distance initiale entre toutes paires de sommets non adjacents. Ensuite, nous étudions et démontrons plusieurs propriétés pour cette classe de graphes. En particulier, après avoir introduit une famille de partitions spécifiques, nous montrons les relations entre certains éléments de cette famille et leur caractère hypotriangulé. De plus, grâce à ces partitions, nous caractérisons les graphes hypotriangulés minimum, qui, parmi les graphes hypotriangulés connexes, minimisent le nombre d'arêtes pour un nombre de sommets fixés.Dans une deuxième partie, nous étudions la complexité, pour la classe des graphes hypotriangulés, de problèmes difficiles dans le cas général. Nous montrons d'abord que les problèmes classiques de cycle hamiltonien, coloration, clique maximum et stable maximum restent NP-difficiles pour cette classe de graphes. Ensuite, nous nous intéressons à des problèmes de modification de graphes, pour lesquels il s'agit de déterminer le nombre minimal d'arêtes à ajouter ou supprimer à un graphe pour obtenir un graphe hypotriangulé : nous montrons la complexité de ces problèmes pour plusieurs classes de graphes. / In this thesis, we define a new class of graphs : the hypochordal graphs. These graphs satisfy that for any path of length two, there exists a chord or another path of length two between its two endpoints. This class can represent robust networks. Indeed, we show that in such graphs, in the case of an edge or a vertex deletion, the distance beween any pair of nonadjacent vertices remains unchanged. Then, we study several properties for this class of graphs. Especially, after introducing a family of specific partitions, we show the relations between some of these partitions and hypochordality. Moreover, thanks to these partitions, we characterise minimum hypochordal graph, that are, among connected hypochordal graphs, those that minimise the number of edges for a given number of vertices. In a second part, we study the complexity, for hypochordal graphs, of problems that are NP-hard in the general case. We first show that the classical problems of hamiltonian cycle, colouring, maximum clique and maximum stable set remain NP-hard for this class of graphs. Then, we analyse graph modification problems : deciding the minimal number of edges to add or delete from a graph, in order to obtain an hypochordal graph. We study the complexity of these problems for sevaral classes of graphs.

We will introduce three new classes of graphs; namely bipartite dot product graphs, probe dot product graphs, and combinatorial orthogonal graphs. All of these representations were inspired by a vector representation known as a dot product representation. Given a bipartite graph G = (X, Y, E), the bipartite dot product representation of G is a function ƒ : X ∪ Y → Rk and a positive threshold t such that for any κ ∈ Χ and γ ∈ Υ , κγ ∈ ε if and only if f(κ) · f(γ) ≥ t. The minimum k such that a bipartite dot product representation exists for G is the bipartite dot product dimension of G, denoted bdp(G). We will show that such representations exist for all bipartite graphs as well as give an upper bound for the bipartite dot product dimension of any graph. We will also characterize the bipartite graphs of bipartite dot product dimension 1 by their forbidden subgraphs. An undirected graph G = (V, E) is a probe C graph if its vertex set can be parti-tioned into two sets, N (nonprobes) and P (probes) where N is independent and there exists E' ⊆ N × N such that G' = (V, E ∪ E) is a C graph. In this dissertation we introduce probe k-dot product graphs and characterize (at least partially) probe 1-dot product graphs in terms of forbidden subgraphs and certain 2-SAT formulas. These characterizations are given for the very different circumstances: when the partition into probes and nonprobes is given, and when the partition is not given. Vectors κ = (κ1, κ2, . . . , κn)T and γ = (γ1, γ2, . . . , γn)T are combinatorially orthogonal if |{i : κiγi = 0}| ≠ 1. An undirected graph G = (V, E) is a combinatorial orthogonal graph if there exists ƒ : V → Rn for some n ∈ Ν such that for any u, υ &Isin; V , uv ∉ E iff ƒ(u) and ƒ(v) are combinatorially orthogonal. These representations can also be limited to a mapping g : V → {0, 1}n such that for any u,v ∈ V , uv ∉ E iff g(u) · g(v) = 1. We will show that every graph has a combinatorial orthogonal representation. We will also state the minimum dimension necessary to generate such a representation for specific classes of graphs.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第18621号 / 情博第545号 / 新制||情||96(附属図書館) / 31521 / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 永持 仁, 教授 太田 快人, 教授 髙橋 豊 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

In this thesis, we consider packings and coverings of various complete graphs with the 4-cycle with a pendant edge. We consider both restricted and unrestricted coverings. Necessary and sufficient conditions are given for such structures for (1) complete graphs Kv, (2) complete bipartite graphs Km,n, and (3) complete graphs with a hole K(v,w).

The personalization of online advertising has now become a necessity for marketing agencies. The tracking technologies such as third-party cookies gives advertisers the ability to recognize internet users across different websites, to understand their behavior and to assess their needs and their tastes. The amount of created data and interactions leads to the creation of a large cross-device identity graph that links different identifiers such as emails to different devices used on different networks. Over time, strongly connected components appear in this graph, too large to represent only the identifiers or devices of only one person or household. The aims of this project is to partition these components according to the structure of the graph and the features associated to the edges without separating identifiers used by a same person. Subsequent to this, the size reduction of these components leads to the isolation of individuals and the identifiers associated to them. This thesis presents the design of a bipartite graph from the available data, the implementation of different community detection graphs adapted to this specific case and different validation methods designed to assess the quality of our partition. Different graph metrics are then used to compare the outputs of the algorithms and we will observe how the adaptation of the algorithm to the bipartite case can lead to better results. / Anpassningen av onlineannonsering har nu blivit en nödvändighet för marknadsföringsbyråer. Spårningstekniken som cookies från tredje part ger annonsörer möjlighet att känna igen internetanvändare på olika webbplatser, för att förstå deras beteende och för att bedöma deras behov och deras smak. Mängden skapade data och interaktioner leder till skapandet av en stor identitetsgrafik för flera enheter som länkar olika identifierare, t.ex. e-postmeddelanden till olika enheter som används i olika nätverk. Över tiden visas starkt anslutna komponenter i det här diagrammet, för stora för att endast representera identifierare eller enheter av endast en person eller hushåll. Syftet med detta projekt är att partitionera dessa komponenter enligt grafens struktur och de egenskaper som är knutna till kanterna utan att separera identifierare som används av samma person. Efter detta leder storleksreduktionen av dessa komponenter till isoleringen av individer och de identifierare som är associerade med dem. Denna avhandling presenterar utformningen av en bifogad graf från tillgängliga data, genomförandet av olika samhällsdetekteringskurvor anpassade till detta specifika fall och olika valideringsmetoder som är utformade för att bedöma kvaliteten på vår partition. Olika grafvärden används då för att jämföra algoritmens utgångar och vi kommer att observera hur anpassningen av algoritmen till tvåpartsfallet kan leda till bättre resultat.

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

Determining the chromatic number of the partition graph P(33) poses a considerable challenge. We can bound it to 4 ≤ χ(P(33)) ≤ 6, with exhaustive search confirming χ(P(33)) = 6. A potential mathematical proof strategy for this equality involves identifying a Z2-invariant S4 with non-trivial homology in the box complex of the partition graph P(33), namely Bedge(︁P(33))︁, and applying the Borsuk-Ulam theorem to compute its Z2-index. This provides a robust topological lower bound for the chromatic number of P(33), termed the Lovász bound. We have verified the absence of such an S4 within certain sections of Bedge(︁P(33))︁. We also validated this approach through a case study on the Petersen graph. This thesis offers a thorough examination of various topological lower bounds for a graph’s chromatic number, complete with proofs and examples. We demonstrate instances where these lower bounds converge to a single value and others where they diverge significantly from a graph’s actual chromatic number. We also classify all vertex pairs, triples, and quadruples of P(33) into unique equivalence classes, facilitating the derivation of all maximal complete bipartite subgraphs. This classification informs the construction of all simplices of Bedge(︁P(33)). Following a detailed and technical exploration, we uncover both the maximal size of the pairwise intersections of its maximal simplices and their underlying structure. Our study proposes an algorithm for building the box complex of the partition graph P(33) using our method of identifying maximal complete bipartite subgraphs. This reduces time complexity to O(n3), marking a substantial enhancement over brute-force techniques. Lastly, we apply discrete Morse theory to construct a simplicial complex homotopy equivalent to the box complex of P(33), using two methods: elementary collapses and the determination of a discrete Morse function on the box complex. This process reduces the dimension of the box complex from 35 to 12, streamlining future calculations of the Z2-index and the Lovász bound.