nondirected

Nondirected is a term sometimes used as a synonym for undirected, especially in graph theory. In a nondirected (undirected) graph, edges have no orientation; an edge simply connects two vertices. By contrast, directed graphs use arrows to indicate direction.

Formally, a nondirected graph G consists of a set V of vertices and a set E of edges, where each edge is an unordered pair {u, v} of vertices from V. Some conventions allow loops, where an edge connects a vertex to itself. The key property is that the incidence relation is symmetric: if u is connected to v by an edge, then v is connected to u by the same edge.

Several properties arise from this symmetry. The adjacency relation is symmetric, and the adjacency matrix of a nondirected graph is symmetric. The degree deg(v) of a vertex v counts incident edges, with a loop contributing two to the degree under standard conventions. Paths and cycles in nondirected graphs are sequences of distinct or repeated vertices where consecutive vertices are joined by edges; the existence of a path between two vertices defines connectivity.

Common variants include simple graphs (no parallel edges or loops), multigraphs (parallel edges allowed), and weighted graphs (edges carry weights). Algorithms such as breadth-first search and depth-first search operate on nondirected graphs and underpin many applications, including detection of connected components, shortest paths in unweighted graphs, and graph traversals.

Applications often model mutual relationships or environments without inherent direction, such as social networks with bidirectional ties, undirected road networks (where two-way streets exist), and various network design problems. The term nondirected emphasizes the lack of orientation, with undirected and nondirected used interchangeably in many contexts.