Connected graph

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This article defines a property that can be evaluated to true/false for any undirected graph, and is invariant under graph isomorphism. Note that the term "undirected graph" as used here does not allow for loops or parallel edges, so there can be at most one edge between two distinct vertices, the edge is completely described by the vertices it joins, and there can be no edge from a vertex to itself.
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An undirected graph is termed connected if it satisfies the following equivalent conditions:

  1. Given any two distinct vertices of the graph, there is a path from one vertex to the other.
  2. It contains a subgraph on the entire vertex set that is a tree.
  3. The geometric realization of the graph is connected as a topological space.

Typically, we do not use the term connected or disconnected for a graph with no vertices.