Philosophiae Doctor - PhD / In this thesis, we describe linear codes over prime fields obtained from incidence designs of iterated line graphs of complete graphs Li(Kn) where i = 1, 2. In the binary case, results are extended to codes from neighbourhood designs of the line graphs Li+1(Kn) using certain elementary relations. Codes from incidence designs of complete graphs, Kn, and neighbourhood designs of their line graphs, L1(Kn) (the so-called triangular graphs), have been considered elsewhere by others. We consider codes from incidence designs of L1(Kn) and L2(Kn), and neighbourhood designs of L2(Kn) and L3(Kn). In each case, basic parameters of the codes are determined. Further, we introduce a family of vertex-transitive graphs Γn that are embeddable into the strong product L1(Kn)⊠  K2, of triangular graphs and K2, a class which at first sight may seem unnatural but, on closer look, is a repository of graphs rich with combinatorial structures. For instance, unlike most regular graphs considered here and elsewhere that only come with incidence and neighbourhood designs, Γn also has what we have termed as 6-cycle designs. These are designs in which the point set contains vertices of the graph and every block contains vertices of a 6-cycle in the graph. Also, binary codes from incidence matrices of these graphs have other minimum words in addition to incidence vectors of the blocks. In addition, these graphs have induced subgraphs isomorphic to the family Hn of complete porcupines (see Definition 4.11). We describe codes from incidence matrices of Γn and Hn and determine their parameters. / South Africa

Dans cette thèse nous étudions quatre problèmes de théorie des graphes. En particulier,Nous étudions le problème du cycle hamiltonien dans les line graphes, et aussi nous prouvons l’existence de cycles hamiltoniens dans certains sous graphes couvrants d’un line graphe. Notre résultat principal est: Si L(G) est hamiltonien, alors SL(G) est hamiltonien. Grâce à ce résultat nous proposons une conjecture équivalente à des conjectures célèbres. Et nous obtenons deux résultats sur les cycles hamiltoniens disjoints dans les line graphes.Nous considérons alors la bipancyclicité résistante aux pannes des graphes de Cayley engendrés par transposition d’arbres. Nous prouvons que de tels graphes de Cayley excepté le “star graph” ont une bipancyclicité (n − 3)-arête résistante aux pannes.Ensuite nous introduisons la coloration des arcs d’un digraphe sommet distinguant. Nous étudions la relation entre cette notion et la coloration d’arêtes sommet distinguant dans les graphes non orientés. Nous obtenons quelques résultats sur le nombre arc chromatique des graphes orientés (semi-)sommet-distinguant et proposons une conjecture sur ce paramètre. Pour vérifier cette conjecture nous étudions la coloration des arcs d’un digraphe sommet distinguant des graphes orientés réguliers.Finalement nous introduisons la coloration acyclique des arcs d’un graphe orienté. Nous calculons le nombre chromatique acyclique des arcs de quelques familles de graphes orientés et proposons une conjecture sur ce paramètre. Nous considérons les graphes orientés de grande maille et utilisons le Lemme Local de Lovász; d’autre part nous considérons les graphes orientés réguliers aléatoires. Nous prouvons que ces deux classes de graphes vérifient la conjecture. / In this thesis, we study four problems in graph theory, the Hamiltonian cycle problem in line graphs, the edge-fault-tolerant bipancyclicity of Cayley graphs generated by transposition trees, the vertex-distinguishing arc colorings in digraph- s and the acyclic arc coloring in digraphs. The first two problems are the classic problem on the cycles in graphs. And the other two arc coloring problems are related to the modern graph theory, in which we use some probabilistic methods. In particular,We first study the Hamiltonian cycle problem in line graphs and find the Hamiltonian cycles in some spanning subgraphs of line graphs SL(G). We prove that: if L(G) is Hamiltonian, then SL(G) is Hamiltonian. Due to this, we propose a conjecture, which is equivalent to some well-known conjectures. And we get two results about the edge-disjoint Hamiltonian cycles in line graphs.Then, we consider the edge-fault-tolerant bipancyclicity of Cayley graphs generated by transposition trees. And we prove that the Cayley graph generated by transposition tree is (n − 3)-edge-fault-tolerant bipancyclic if it is not a star graph.Later, we introduce the vertex-distinguishing arc coloring in digraphs. We study the relationship between the vertex-distinguishing edge coloring in undirected graphs and the vertex-distinguishing arc coloring in digraphs. And we get some results on the (semi-) vertex-distinguishing arc chromatic number for digraphs and also propose a conjecture about it. To verify the conjecture we study the vertex-distinguishing arc coloring for regular digraphs.Finally, we introduce the acyclic arc coloring in digraphs. We calculate the acyclic arc chromatic number for some digraph families and propose a conjecture on the acyclic arc chromatic number. Then we consider the digraphs with high girth by using the Lovász Local Lemma and we also consider the random regular digraphs. And the results of the digraphs with high girth and the random regular digraphs verify the conjecture.

Philosophiae Doctor - PhD / In this thesis, we describe linear codes over prime fields obtained from incidence designs of iterated line graphs of complete graphs Li(Kn) where i = 1,2. In the binary case, results are extended to codes from neighbourhood designs of the line graphs Li+l(Kn) using certain elementary relations. Codes from incidence designs of complete graphs, Kn' and neighbourhood designs of their line graphs, £1(Kn) (the so-called triangular graphs), have been considered elsewhere by others. We consider codes from incidence designs of Ll(Kn) and L2(Kn), and neighbourhood designs of L2(Kn) and L3(Kn). In each case, the basic parameters of the codes are determined. Further, we introduce a family of vertex-transitive graphs Rn that are embeddable into the strong product Ll(Kn) ~ K2' of triangular graphs and K2' a class that at first sight may seem unnatural but, on closer look, is a repository of graphs rich with combinatorial structures. For instance, unlike most regular graphs considered here and elsewhere that only come with incidence and neighbourhood designs, Rn also has what we have termed as 6-cycle designs. These are designs in which the point set contains vertices of the graph and every block contains vertices of a 6-cycle in the graph. Also, binary codes from incidence matrices of these graphs have other minimum words in addition to incidence vectors of the blocks. In addition, these graphs have induced subgraphs isomorphic to the family Hn of complete porcupines (see Definition 4.11). We describe codes from incidence matrices of Rn and Hn and determine their parameters. The discussion is concluded with a look at complements of Rn and Hn, respectively denoted by Rn and Hn. Among others, the complements rn are contained in the union of the categorical product Ll(Kn) x Kn' and the categorical product £1(Kn) x Kn (where £1(Kn) is the complement of the iii triangular graph £1(Kn)). As with the other graphs, we have also considered codes from the span of incidence matrices of Rn and Hn and determined some of their properties. In each case, automorphisms of the graphs, designs and codes have been determined. For the codes from incidence designs of triangular graphs, embeddings of Ll(Kn) x K2 and complements of complete porcupines, we have exhibited permutation decoding sets (PD-sets) for correcting up to terrors where t is the full error-correcting capacity of the codes. For the remaining codes, we have only been able to determine PD-sets for which it is possible to correct a fraction of t-errors (partial permutation decoding). For these codes, we have also determined the number of errors that can be corrected by permutation decoding in the worst-case.

This work deals with problems of easy web graph generate. Project implement user-friendly interface for Object-Oriented Graph creating library pChart.

The flight search engine is a technology used in the air travel industry. It allows the traveler to search and book for the best flight options, such as the combination of flights while keeping the best services, options, and price. The computation for a flight search query can be very intensive given its parameters and complexity. The project goal is to predict the flight search queries computation cost for a new flight search engine product when dealing with parameters change and optimizations. The problem of flight search cost prediction is a regression problem. We propose to solve the problem by delimiting the problem based on its business logic and meaning. Our problem has data defined as a graph, which is why we have chosen Graph Neural Network. We have investigated multiple pretraining strategies for the evaluation of node embedding concerning a realworld regression task, including using a line graph for the training. The embeddings are used for downstream regression tasks. Our work is based on some stateoftheart Machine Learning, Deep Learning, and Graph Neural Network methods. We conclude that for some business use cases, the predictions are suitable for production use. In addition, the prediction of tree ensemble boosting methods produces negatives predictions which further degrade the R2 score by 4% because of the business meaning. The Deep Neural Network outperformed the most performing Machine Learning methods by 8% to 12% of R2 score. The Deep Neural Network also outperformed Deep Neural Network with pretrained node embedding from the Graph Neural Network methods by 11% to 17% R2 score. The Deep Neural Network achieved 93%, 81%, and 63% R2 score for each task with increasing difficulty. The training time range from 1 hour for Machine Learning models, 2 to 10 hours for Deep Learning models, and 8 to 24 hours for Deep Learning model for tabular data trained end to end with Graph Neural Network layers. The inference time is around 15 minutes. Finally, we found that using Graph Neural Network for the node regression task does not outperform Deep Neural Network. / Flygsökmotor är en teknik som används inom flygresebranschen. Den gör det möjligt för resenären att söka och boka de bästa flygalternativen, t.ex. kombinationer av flygningar med bästa service, alternativ och pris. Beräkningen av en flygsökning kan vara mycket intensiv med tanke på dess parametrar och komplexitet. Projektets mål är att förutsäga beräkningskostnaden för flygsökfrågor för en ny produkt för flygsökmotor när parametrar ändras och optimeringar görs. Problemet med att förutsäga kostnaderna för flygsökning är ett regressionsproblem. Vi föreslår att man löser problemet genom att avgränsa det utifrån dess affärslogik och innebörd. Vårt problem har data som definieras som en graf, vilket är anledningen till att vi har valt Graph Neural Network. Vi har undersökt flera förträningsstrategier för utvärdering av nodinbäddning när det gäller en regressionsuppgift från den verkliga världen, bland annat genom att använda ett linjediagram för träningen. Inbäddningarna används för regressionsuppgifter i efterföljande led. Vårt arbete bygger på några toppmoderna metoder för maskininlärning, djupinlärning och grafiska neurala nätverk. Vi drar slutsatsen att förutsägelserna är lämpliga för produktionsanvändning i vissa Vi drar slutsatsen att förutsägelserna är lämpliga för produktionsanvändning i vissa fall. Dessutom ger förutsägelserna från trädens ensemble av boostingmetoder negativa förutsägelser som ytterligare försämrar R2poängen med 4% på grund av affärsmässiga betydelser. Deep Neural Network överträffade de mest effektiva metoderna för maskininlärning med 812% av R2poängen. Det djupa neurala nätverket överträffade också det djupa neurala nätverket med förtränad node embedding från metoderna för grafiska neurala nätverk med 11 till 17% av R2poängen. Deep Neural Network uppnådde 93, 81 och 63% R2poäng för varje uppgift med stigande svårighetsgrad. Träningstiden varierar från 1 timme för maskininlärningsmodeller, 2 till 10 timmar för djupinlärningsmodeller och 8 till 24 timmar för djupinlärningsmodeller för tabelldata som tränats från början till slut med grafiska neurala nätverkslager. Inferenstiden är cirka 15 minuter. Slutligen fann vi  att användningen av Graph Neural Network för uppgiften om regression av noder inte överträffar Deep Neural Network.