Question

The Girth Of A Graph Is The Length Of Its Shortest Cycle. Show That If G Is A Graph Of Diameter D And Girth 2d+1, Then G Is irregular

Answer 1

To show that if a graph G has a diameter D and girth 2D+1, then G is irregular, we can use the fact that the length of the shortest cycle in the graph is equal to its girth.

Step-by-step explanation:

To show that if a graph G has a diameter D and girth 2D+1, then G is irregular, we can use the fact that the length of the shortest cycle in the graph is equal to its girth. Since the girth of G is 2D+1, it means that the shortest cycle in G has length 2D+1.

Now, if G is regular, then all vertices in G must have the same degree. However, the shortest cycle in G has an odd number of vertices (2D+1), which means that each vertex in the cycle has an odd degree. But if all vertices in the graph have the same degree and all have an odd degree, then the sum of the degrees of all vertices must be odd, which is not possible. Therefore, G cannot be regular and must be irregular.