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Consider the tree given below:

Using the property of eccentricity of a vertex, find every vertex that is the centre of the given tree:

1. d & h
2. c & k
3. g, b, c, h, i, m
4. c & h
in DS
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Eccentricity of Vertex:-The maximum distance between a vertex to all other vertices is considered as the eccentricity of vertex denoted by e(V).The distance from a particular vertex to all other vertices in the graph is taken and among those distances, the eccentricity is the highest of distances.

For this graph

e(a)=5                                e(b)=4                                  e(c)=3

e(d)=4                                e(e)=5                                  e(f)=5

e(g)=4                                e(h)=3                                  e(i)=4

e(j)=5                                 e(k)=4                                  e(l)=5

e(m)=4                               e(n)=5

Radius:-The minimum eccentricity from all the vertices is considered as the radius of the Graph G denoted by r(G). The minimum among all the maximum distances between a vertex to all other vertices is considered as the radius of the Graph G.

Here

r(c)=3 and r(h)=3 which is the minimum eccentricity for  c and h.

Center:- If the eccentricity of a graph is equal to its radius, then it is known as the central point of the graph. If

e(V) = r(V),

then ‘V’ is the central point of the Graph ’G’.

Here

e(c)=r(c)=3                         and e(h)=r(h)=3

Hence,both c and h are center of tree.

Hence,Option(D)c & h is the correct choice.

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by Boss (40.8k points)
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When finding distance..it should be minimum...

Since between two vertices many paths are possible we only select which have minimum weight..

Among all these minimum distances from source to all other vertices maximum taken as eccentricity..
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The distance from a particular vertex to all other vertices in the graph is taken and among those distances, the eccentricity is the highest of distances.

It is number of edges in a shortest path between Vertex U and Vertex V. If there are multiple paths connecting two vertices, then the shortest path is considered as the distance between the two vertices.

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