eccentricity of all the vertices in the strong product of two given graphs

Question

I am trying to prove a result, where I have to show that the strong product of $K_{2}$ and $P_n$, n>2, is not self-centered graph, that is, eccentricity is not equal for all the vertices. the idea which i am having is that in the product of graphs, distance between two vertices $g_{hi}$ and $g_{jk}$ is equal to the sum of $d_V$($u_h$,$u_h$) and $d_V$($v_i$,$v_k$), and thus $e_{U \times V}$($g_{ij}$) $=$ $e_Uu_i$ + $e_Vv_j$ where V and U are given graphs. i am using this idea that eccentricty of vertices in a path keep changing as we increase the number of vertices.

Answer 1

in the strong product, the distance $d(x,y)$ for vertices with coordinates $x=(x_1,..x_n)$ and $y=(y_1,..y_n)$ is the maximum of the distances $d(x_i,y_i)$ over each factor. The sum holds for Cartesian products only.