In each case, either draw a graph G with the given graph parameters, or prove th

Question

In each case, either draw a graph G with the given graph parameters, or prove that no such graph exists. (a) (G) = 2, (G) = 3, and diam(G) = 4. (b) rad(G) = 2, (G) = 3, and (G) = 4. (c) (G) = 3, (G) = 3, diam(G) = 3, and rad(G) = 3. In each case, either draw a graph G with the given graph parameters, or prove that no such graph exists. (a) (G) = 2, (G) = 3, and diam(G) = 4. (b) rad(G) = 2, (G) = 3, and (G) = 4. (c) (G) = 3, (G) = 3, diam(G) = 3, and rad(G) = 3. In each case, either draw a graph G with the given graph parameters, or prove that no such graph exists. (a) (G) = 2, (G) = 3, and diam(G) = 4. (b) rad(G) = 2, (G) = 3, and (G) = 4. (c) (G) = 3, (G) = 3, diam(G) = 3, and rad(G) = 3.

Explanation / Answer

(A):
(G) denote for connectivity(vertex) of a graph G and (G) denote edge-connectivity of graph G.
Also if, (G) (G), the graphs are all connected. Therefore, for every graph G, (G) (G)
In our case, (G) =2 and (G)=3, hence, (G) (G). Hence, graph exists.


(C):
For any connected graph G, rad(G) diam(G) 2 rad(G). This is verified in the question.

The maximum degree of a graph G, denoted by (G), is defined to be:

(G) = max{deg(v) | v V (G)}.

Similarly, the minimum degree of a graph G, denoted by (G), is defined to be:

(G) = min{deg(v) | v V (G)}.

For graph to exist, (G) < (G)

Since this is not happening in the question, hence graph doesn't exist.

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