What is the diameter of $C_n$ and $P_n$ I need to find for each $3\le n\in N $ what is $C_n$ Diameter and his Complement graph Diameter
and the same for each $1\le n\in N $ what is $P_n$ Diameter and his Complement graph Diameter.
Here is what i done:
I found out for $C_n$ that its $\lfloor {n\over 2} \rfloor$  and his Complement $\lfloor {n-2\over 2} \rfloor$
I think that $P_n$ is just $n$ but i cant find his Complement Diameter
is that correct? and how can i find the Complement Diameter
 A: You are right about Cn diameter, it is $\lfloor \frac{n}{2} \rfloor$. For Cnc, it varies, I have'nt found a generalised answer for this, but it is not what you have mentioned. Take example of C4, it's complement is not even a connected graph.
For n=3, Pnc is not connected.
For n>=4
You probably made a mistake in counting the diameter of Pn. It is n-1.The complement of Pn is always 2. (How?)
There are 2 leafs say $u,v$ in Pn, and rest of the vertices are of degree 2. Now, when we take the complement, the leaves would be connected to every other vertex except to which it was connected earlier say $w$. Now, to check the eccentricity of $u$, it is obvious that its distance to every vertex other than $w$ is 1, but to reach $w$, it needs to go to another vertex ($v$), thus D($u,w$) = 2. Which implies e($u$)=e($v$)=2. Now, lets take a non-leaf $w$, it's distance to every vertex except to which it was adjacent (say $x$ and $y$) in Pn is 1.  But to reach $x$ or $y$, it needs to go to $v$ and then come to $x$, thus D($w,x$) = 2 = D($w,y$). Therefore, e($w$) = 2.
Thus eccentricity of every vertex in Pnc is 2 and hence diamneter of Pnc is 2.
