Littérature scientifique sur le sujet « Graphe quasi-Bipartite »

In this paper, we study some graph-theoretic properties about the zero-divisor graph [Formula: see text] of a finite quasi-ordered set [Formula: see text] with a least element 0 and its line graph [Formula: see text]. First, we offer a method to find all the minimal prime ideals of a quasi-ordered set. Especially, this method is applicable for a partially ordered set. Then, we completely characterize the diameter and girth of [Formula: see text] by the minimal prime ideals of [Formula: see text]. Besides, we perfectly classify all finite quasi-ordered sets whose zero-divisor graphs are complete graphs, star graphs, complete bipartite graphs, complete [Formula: see text]-partite graphs. We also investigate the planarity of [Formula: see text]. Finally, we obtain the characterization for the line graph [Formula: see text] in terms of its diameter, girth and planarity.

Let M be a module over a commutative ring R. In this paper, we continue our study about the quasi-Zariski topology-graph G(?*T) which was introduced in (On the graph of modules over commutative rings, Rocky Mountain J. Math. 46(3) (2016), 1-19). For a non-empty subset T of Spec(M), we obtain useful characterizations for those modules M for which G(?*T) is a bipartite graph. Also, we prove that if G(?*T) is a tree, then G(?*T) is a star graph. Moreover, we study coloring of quasi-Zariski topology-graphs and investigate the interplay between ?(G(?+T)) and ?(G(?+T)).

This work is aimed to introduce a new topology on a graph, namely the degree topology. This topology is defined by the degree of the vertices of the graphs. We find the degree topology for certain types of graphs and determine their types. The degree topology for the complete graph is an indiscrete topology. While The degree topology is generated by a complete bipartite graph with is a quasi-discrete topology. In addition, a new property is initiated namely set- space and discussed the link between it and space. We verify that every degree topology is a set- space.

In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2–factor 𝒬 via (quasi) spanning trees of faces in G/𝒬 and study the algorithmic complexity of finding such (quasi) spanning trees of faces. Moreover, we show that if Barnette’s Conjecture is false, then hamiltonicity in 3–connected planar cubic bipartite graphs is an NP-complete problem.

Let G be a connected non-regular non-bipartite graph whose adjacency matrix has spectrum p,?(k),?(l), where k,l ? N and p > ? > ?. We show that if ? is non-main then ?(G) ? 1 + ? - ??, with equality if and only if G is of one of three types, derived from a strongly regular graph, a symmetric design or a quasi-symmetric design (with appropriate parameters in each case).

LetGbe an undirected simple graph of ordern. LetA(G)be the adjacency matrix ofG, and letμ1(G)≤μ2(G)≤⋯≤μn(G)be its eigenvalues. The energy ofGis defined asℰ(G)=∑i=1n‍|μi(G)|. Denote byGBPTa bipartite graph. In this paper, we establish the sufficient conditions forGhaving a Hamiltonian path or cycle or to be Hamilton-connected in terms of the energy of the complement ofG, and give the sufficient condition forGBPThaving a Hamiltonian cycle in terms of the energy of the quasi-complement ofGBPT.

Dans cette thèse, nous étudions les polyèdres total dual box-intègraux (box-TDI) associés à plusieurs problèmes et matrices totalement équimodulaires. De plus, nous étudions la complexité de certaines questions fondamentales liées à ces polyèdres. Nous commençons par considérer les matrices totalement équimodulaires, qui sont des matrices telles que, pour chaque sous-ensemble de lignes linéairement indépendantes, toutes les sous-matrices maximales non-singulières ont le même déterminant en valeur absolue. Malgré leurs similitudes avec les matrices totalement unimodulaires, nous mettons en évidence plusieurs différences, même dans le cas des matrices d'incidence et d'adjacence des graphes. Comme on le sait, la matrice d'incidence d'un graphe donné est totalement unimodulaire si et seulement si le graphe est biparti. Cependant, la totale équimodularité d'une matrice d'incidence dépend du fait que nous considérons la représentation sommet-arête ou arête-sommet. Nous fournissons des caractérisations pour les deux cas. En conséquence, nous prouvons que reconnaître si un polyèdre donné est box-TDI est un problème co-NP-complet. La caractérisation de la totale unimodularité ou de la totale équimodularité de la matrice d'adjacence d'un graphe biparti donné reste non résolue, alors que nous avons résolu le problème correspondant dans le cas de la totale équimodularité lorsque le graphe est non-biparti. Dans une dernière partie, nous caractérisons les graphes pour lesquels le polytope des couplages parfaits (PMP) est décrit par des inégalités triviales et des inégalités correspondant à des coupes serrées. Les coupes serrées sont définies comme des coupes qui partagent précisément une arête avec chaque couplage parfait. Nous prouvons ensuite que tout graphe pour lequel le PMP correspondant est box-TDI appartient à cette classe. En conséquence, reconnaître si le PMP est box-TDI est un problème résoluble en temps polynomial. Cependant, nous fournissons plusieurs contre-exemples montrant que cette classe de graphes ne garantit pas la box-TDIness du PMP. Enfin, nous présentons des conditions nécessaires pour un polytope de couverture des arêtes pour être box-TDI et caractérisons quand le polytope des couplages extensibles est box-TDI, qui est l'enveloppe convex des couplages inclus dans un couplage parfait
In this thesis, we study box-totally dual integral (box-TDI) polyhedra associated with severalproblems and totally equimodular matrices. Moreover, we study the complexity of some funda-mental questions related to them.We start by considering totally equimodular matrices, which are matrices such that, forevery subset of linearly independent rows, all nonsingular maximal submatrices have the samedeterminant in absolute value. Despite their similarities with totally unimodular matrices, wehighlight several differences, even in the case of incidence and adjacency matrices of graphs.As is well-known, the incidence matrix of a given graph is totally unimodular if and only if thegraph is bipartite. However, the total equimodularity of an incidence matrix depends on whetherwe consider the vertex-edge or the edge-vertex representation. We provide characterizations forboth cases. As a consequence, we prove that recognizing whether a given polyhedron is box-TDIis a co-NP-complete problem.Characterizing the total unimodularity or total equimodularity of the adjacency matrix of agiven bipartite graph remains unsolved, while we solved the corresponding problem in the case oftotal equimodularity when the graph is nonbipartite.In a later part of this work, we characterize the graphs for which the perfect matching polytope(PMP) is described by trivial inequalities and the inequalities corresponding to tight cuts. Tightcuts are defined as cuts that share precisely one edge with each perfect matching. We thenprove that any graph for which the corresponding PMP is box-TDI belongs to this class. Asa consequence, it turns out that recognizing whether the PMP is box-TDI is a polynomial-timeproblem. However, we provide several counterexamples showing that this class of graphs does notguarantee the box-TDIness of the PMP.Lastly, we present necessary conditions for the box-TDIness of the edge cover polytope andcharacterize the box-TDIness of the extendable matching polytope, which is the convex hull ofthe matchings included in a perfect matching