The word "connected" speaks to the most basic structural properties of networks. It is arguably both the most important and the most overused term in the network vocabulary.
Important and proper uses of "connected":
Connected has several distinct formal definitions, each of which is important. We introduce three of them here and elaborate the details later.
- A path connects its origin and destination nodes
- Two nodes are connected if there is at least one path that connects them
- A graph is connected if each of its nodes is connected to all the others
Examples:
- Nodes 6 and 2 are connected in graph G1 below
- Nodes a and b are not connected in graph G2 below
- Graph G1 is connected; graph G2 is not connected
 |  |
Graph G1 = (V1,E1) | Graph G2 = (V2,E2) |
Common sloppy uses of "connected":
Sometimes "connected" is used informally when another term would be more clear. For example, avoid using "connected" when one of these unambiguous words would apply:
- Joined
- Adjacent
- Incident
