Undirected graph in the context of Girth (graph theory)

A directed tree, oriented tree, polytree, or singly connected network is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest.

A semantic network, or frame network is a knowledge base that represents semantic relations between concepts in a network. This is often used as a form of knowledge representation. It is a directed or undirected graph consisting of vertices, which represent concepts, and edges, which represent semantic relations between concepts, mapping or connecting semantic fields. A semantic network may be instantiated as, for example, a graph database or a concept map. Typical standardized semantic networks are expressed as semantic triples.

Semantic networks are used in natural language processing applications such as semantic parsing and word-sense disambiguation. Semantic networks can also be used as a method to analyze large texts and identify the main themes and topics (e.g., of social media posts), to reveal biases (e.g., in news coverage), or even to map an entire research field.

In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the left child and the right child. That is, it is a k-ary tree where k = 2. A recursive definition using set theory is that a binary tree is a triple (L, S, R), where L and R are binary trees or the empty set and S is a singleton (a single–element set) containing the root.

From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence, a term which appears in some early programming books before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of binary tree to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted.

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).

Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose.

The Hamiltonian path problem is a topic discussed in the fields of complexity theory and graph theory. It decides if a directed or undirected graph, G, contains a Hamiltonian path, a path that visits every vertex in the graph exactly once. The problem may specify the start and end of the path, in which case the starting vertex s and ending vertex t must be identified.

The Hamiltonian cycle problem is similar to the Hamiltonian path problem, except it asks if a given graph contains a Hamiltonian cycle. This problem may also specify the start of the cycle. The Hamiltonian cycle problem is a special case of the travelling salesman problem, obtained by setting the distance between two cities to one if they are adjacent and two otherwise, and verifying that the total distance travelled is equal to n. If so, the route is a Hamiltonian cycle.

In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).

The name line graph comes from a paper by Harary & Norman (1960) although both Whitney (1932) and Krausz (1943) used the construction before this. Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the θ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph.

In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs. The analogue concept for polytopes of general dimension are the polytopal graphs.

In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components.

The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component, one component that is significantly larger than the others; and of a percolation threshold, an edge probability above which a giant component exists and below which it does not.

In the mathematical field of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree (see about spanning forests below). If all of the edges of G are also edges of a spanning tree T of G, then G is a tree and is identical to T (that is, a tree has a unique spanning tree and it is itself).

In graph theory, a random geometric graph (RGG) is the mathematically simplest spatial network, namely an undirected graph constructed by randomly placing N nodes in some metric space (according to a specified probability distribution) and connecting two nodes by a link if and only if their distance is in a given range, e.g. smaller than a certain neighborhood radius, r.

Random geometric graphs resemble real human social networks in a number of ways. For instance, they spontaneously demonstrate community structure - clusters of nodes with high modularity. Other random graph generation algorithms, such as those generated using the Erdős–Rényi model or Barabási–Albert (BA) model do not create this type of structure. Additionally, random geometric graphs display degree assortativity according to their spatial dimension: "popular" nodes (those with many links) are particularly likely to be linked to other popular nodes.