Can some one help me Find the diameter and radius of complete graph with n vertices, I know how to do it for complete graph with small number of vertices but can generalize to the one with n vertices.
[Math] Diameter and radius of complete graph
graph theory
Related Solutions
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.
There are such graphs.
I'll give you a hint.
Hint 1. Which graph is the first one you should think of when trying to find a counterexample or disprove something?
Hint 2. 10 vertices, 3-regular.
Best Answer
Let $G=(V,E)$ be an (unweighted) graph. The eccentricity of a vertex $v\in V$ is $\varepsilon (v)=\max_{u\in V} d(u,v)$. The radius $r$ of $G$ is the minimum eccentricity of the vertices, i.e. $r = \min_{v\in V}\varepsilon(v) = \min_{v\in V}\max_{u\in V}d(u,v)$. The diameter $d$ of $G$ is the maximum eccentricity of the vertices, that is, $d = \max_{v\in V}\varepsilon(v)$.
If $G$ is a complete graph, then $uv\in E$ for all $u,v\in V$, and so $\varepsilon(v)=1$ for all $v\in V$. It follows immediately that the radius and diameter are both one.