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sunnyceej
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prove that for any graph G, kappa (G) ≤delta (G).
sunnyceej said:prove that for any graph G, kappa (G) ≤delta (G).
The minimum degree of a graph is the smallest number of edges connected to a single vertex in the graph.
Connectivity in a graph refers to the minimum number of edges that need to be removed in order to disconnect the graph into two or more separate components.
No, the connectivity of a graph can never be greater than the minimum degree. This is because the minimum degree represents the smallest number of edges connected to a vertex, and removing these edges would disconnect the graph.
To prove this, you can use the Handshaking Lemma, which states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges. Since the connectivity is the minimum number of edges needed to disconnect a graph, it must be less than or equal to the sum of the degrees, which is always less than the minimum degree.
Yes, this statement is true for all types of graphs, including directed and weighted graphs. As long as the graph has a minimum degree and connectivity, this statement holds true.