Radius, diameter and center of graph

Solution 1:

The distance is defined as the number of edges on the shortest path between the vertices. For example, adjacent vertices have a distance of $1$.

In your graph, it might be helpful to explicitly enumerate the eccentricity of each vertex. It is not too difficult to eye-ball the eccentricity for each vertex. I have labelled your graph below with the vertex eccentricities

You can see that in this graph, the larger eccentricities occur at the sides of the graph, with the largest (the diameter) being $6$. The smallest eccentricity occurs at the central vertex with an eccentricity of $3$. This is your radius. Your center consists of all the vertices which have eccentricity equal to the radius, in this case $3$. For this graph, there is only a single such vertex, so your center in the single vertex labelled $3$ in the graph.

You can see that the name center really is quite aptly named. The vertices of the center minimizes the maximum distance to any vertex of the graph and in this sense, really are the most central vertices in the graph.