Proof: Degree Sum Condition for Connected Graphs | Connected Graphs, Nonadjacent Vertices
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Graph Theory course: • Graph Theory
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If every pair of nonadjacent vertices in a graph has a degree sum greater than or equal to one less than the number of vertices in the graph, then the graph is connected and has a diameter less than or equal to 2, and we will prove this theorem in today's video graph theory lesson!
This is a sufficient condition for a graph to be connected, meaning if it is true about some graph G, then G is connected, but if it is not true about some graph H, we still do not know if H is connected or not, it may or may not be. For example, a path graph on 4 vertices does not fill this condition (it has two non-adjacent end vertices, with degree sum 2, which is not greater than or equal to 4 - 1 = 3).
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