I was studying about graph operation on wiki. Corona product of graphs $G_1$ and $G_2$, is the graph which is the disjoint union of one copy of $G_1$ and $|V_1|$ copies of $G_2$ ($|V_1|$ is the number of vertices of $G_1$) in which each vertex of the copy of $G_1$ is connected to all vertices of a separate copy of $G_2$.
After applying the result on few graphs, I came to this result that the eccentricity of every vertex is not same. For ex in the following example eccentricity is not same for all vertices. Is there any theoretical way by which I can proof the same for any graphs? I tried by taking the contradiction as but no success. Can anybody help me here. I jus need a hint, not the full proof. Thanks for helping me $\ddot\smile$
Am I right in asking this question? Is there any graph for which ecc is same ?
Credit for this graph goes to the user https://math.stackexchange.com/users/26306/dtldarek.
$\ddot\smile$
If $G_1$ has only one vertex, it’s possible for every vertex of the corona product to have the same eccentricity: take $G_2$ to be any complete graph. However, if $|V_1|\ge 2$, you’re right. Pick $v_1\in V_1$ of maximal eccentricity $e$, and let $v_2\in V_1$ be such that $d(v_1,v_2)=e$. Show that the eccentricity of $v_1$ in the corona product is $e+1$. Then pick a vertex $u$ in the copy of $G_2$ attached to $v_1$, and show that the eccentricity of $u$ in the corona product is $e+2$.