The set of regions of a map can be represented more abstractly as an undirected graph that has a vertex for each region and an edge for every pair of regions that share a boundary segment.

A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another ; see graph ( mathematics ) for more detailed definitions and for other variations in the types of graph that are commonly considered.

This causes redundancy in an undirected graph: for example, if vertices A and B are adjacent, A's adjacency list contains B, while B's list contains A. Adjacency queries are faster, at the cost of extra storage space.

It is a directed or undirected graph consisting of vertices, which represent concepts, and edges.

TSP can be modelled as an undirected weighted graph, such that cities are the graph's vertices, paths are the graph's edges, and a path's distance is the edge's length.

In the symmetric TSP, the distance between two cities is the same in each opposite direction, forming an undirected graph.

Given a connected, undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together.

More generally, any undirected graph ( not necessarily connected ) has a minimum spanning forest, which is a union of minimum spanning trees for its connected components.

A path in an undirected graph is a sequence of vertices

In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path.

A tree is an undirected simple graph G that satisfies any of the following equivalent conditions:

A forest is an undirected graph, all of whose connected components are trees ; in other words, the graph consists of a disjoint union of trees.

Equivalently, a forest is an undirected cycle-free graph.

A polytree or oriented tree is a directed graph with at most one undirected path between any two vertices.

In other words, a polytree is a directed acyclic graph for which there are no undirected cycles either.

Cayley's formula is the special case of complete graphs in a more general problem of counting spanning trees in an undirected graph, which is addressed by the matrix tree theorem.

In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a connected weighted undirected graph.

* In graph theory, a complete graph is an undirected graph in which every pair of vertices has exactly one edge connecting them.

Intuitively, an expander is a finite, undirected multigraph in which every subset of the vertices " which is not too large " has a " large " boundary.

Plantinga asserts that the design plan does not require a designer: " it is perhaps possible that evolution ( undirected by God or anyone else ) has somehow furnished us with our design plans ", but the paradigm case of a design plan is like a technological product designed by a human being ( like a radio or a wheel ).

An undirected graph has a cycle if and only if a depth-first search ( DFS ) finds an edge that points to an already-visited vertex ( a back edge ).

However there are many other kinds of directed acyclic graph that are not formed by orienting the edges of an undirected acyclic graph, and every undirected graph has an acyclic orientation, an assignment of a direction for its edges that makes it into a directed acyclic graph.

Identifying the in-place algorithms with L has some interesting implications ; for example, it means that there is a ( rather complex ) in-place algorithm to determine whether a path exists between two nodes in an undirected graph, a problem that requires O ( n ) extra space using typical algorithms such as depth-first search ( a visited bit for each node ).

A hypergraph is a combinatorial structure that, like an undirected graph, has vertices and edges, but in which the edges may be arbitrary sets of vertices rather than having to have exactly two endpoints.

An Eulerian orientation of an undirected graph G is an assignment of a direction to each edge of G such that, at each vertex v, the indegree of v equals the outdegree of v. Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of G and then orienting the edges according to the tour.

* An undirected graph has an Eulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single connected component.

* An undirected graph has an Eulerian trail if and only if at most two vertices have odd degree, and if all of its vertices with nonzero degree belong to a single connected component.

* A directed graph has an Eulerian trail if and only if at most one vertex has ( out-degree ) − ( in-degree ) = 1, at most one vertex has ( in-degree ) − ( out-degree ) = 1, every other vertex has equal in-degree and out-degree, and all of its vertices with nonzero degree belong to a single connected component of the underlying undirected graph.

* The handshaking lemma, proven by Euler in his original paper, showing that any undirected connected graph has an even number of odd-degree vertices

In graph theory an undirected graph G has two kinds of incidence matrices: unoriented and oriented.

* Simple graph, an undirected graph that has no loops and no more than one edge between any two different vertices.

An implication graph must be a skew-symmetric graph, meaning that the undirected graph formed by forgetting the orientations of its edges has a symmetry that takes each variable to its negation and reverses the orientations of all of the edges.

2-satisfiability has also been applied to problems of recognizing undirected graphs that can be partitioned into an independent set and a small number of complete bipartite subgraphs, inferring business relationships among autonomous subsystems of the internet, and reconstruction of evolutionary trees.

In an undirected graph, the relation over the set of vertices of the graph under which v and w are related if and only if they are adjacent forms a symmetric relation.

Conversely, if R is a symmetric relation over a set X, one can interpret it as describing an undirected graph with the elements of X as the vertices and the pairs in R as the edges.

Thus, symmetric relations and undirected graphs are combinatorially equivalent objects.

If the graph is undirected, the adjacency matrix is symmetric.

Let G be a finite simple undirected graph with edge set E. The power set of E becomes a Z < sub > 2 </ sub >- vector space if we take the symmetric difference as addition, identity function as negation, and empty set as zero.

The edges may be directed ( asymmetric ) or undirected ( symmetric ).

The convention followed here is that an adjacent edge counts 1 in the matrix for an undirected graph.

For example the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows ( corresponding to the four vertices, 1-4 ) and 4 columns ( corresponding to the four edges, e1-e4 ):

An oriented incidence matrix of an undirected graph G is the incidence matrix, in the sense of directed graphs, of any orientation of G. That is, in the column of edge e, there is one + 1 in the row corresponding to one vertex of e and one − 1 in the row corresponding to the other vertex of e, and all other rows have 0.