Cycle (graph theory)In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an acyclic graph (or a forest). A directed graph without directed cycles is called a directed acyclic graph. A connected graph without cycles is called a tree.
Bipartite graphIn the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent setsU{\displaystyle U} and V{\displaystyle V}, that is, every edge connects a vertex in U{\displaystyle U} to one in V{\displaystyle V}. Vertex sets U{\displaystyle U} and V{\displaystyle V} are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
Degree (graph theory)In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v{\displaystyle v} is denoted deg(v){\displaystyle \deg(v)} or degv{\displaystyle \deg v}.
Independent set (graph theory)In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S{\displaystyle S} of vertices such that for every two vertices in S{\displaystyle S}, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S{\displaystyle S}. A set is independent if and only if it is a clique in the graph's complement.Vadim G. VizingVadim Georgievich Vizing (Russian: Вади́м Гео́ргиевич Визинг, Ukrainian: Вадим Георгійович Візінг; 25 March 1937 – 23 August 2017) was a Soviet and Ukrainian mathematician known for his contributions to graph theory, and especially for Vizing's theorem stating that the edges of any simple graph with maximum degree Δ can be colored with at most Δ + 1 colors.
Graph theoryArea of discrete mathematicsA graph with 6 vertices and 7 edgesIn mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called arcs, links, or lines).
Path (graph theory)In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.
Edge coloringIn graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring.
MultigraphIn mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge. There are 2 distinct notions of multiple edges: Edges without own identity: The identity of an edge is defined solely by the two nodes it connects.
NovosibirskNovosibirsk is the administrative centre and largest city of Novosibirsk Oblast and the Siberian Federal District in Russia. As of the 2021 census, it had a population of 1,633,595, making it the third-most populous city in Russia after Moscow and Saint Petersburg. Additionally, it is the largest city in North Asia and the most populous Russian city that does not have the status of a federal subject.
Disjoint unionIn mathematics, the disjoint union (or discriminated union) A⊔B{\displaystyle A\sqcup B} of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.
Induced subgraphIn graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges, from the original graph, connecting pairs of vertices in that subset.
Graph (discrete mathematics)In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line).*As an Amazon Associate I earn from qualifying purchases.