If I am looking at a Cartesian product of two graphs $G_1$ and $G_2$ (defined here https://en.wikipedia.org/wiki/Cartesian_product_of_graphs).
I am trying to bound the diameter of the graph $G_1 \square G_2$ based on the diameters of graphs $G_1$ and $G_2$.
Everything that I can find on this is the following: https://www.researchgate.net/publication/263799751_THE_DIAMETER_VARIABILITY_OF_THE_CARTESIAN_PRODUCT_OF_GRAPHS, but by the looks of it, it's not answering my question. I am looking to bound the diameter from below.
By my intuition, it seems like $\text{diam}(G_1\square G_2)\ge \text{diam}(G_1)+\text{diam}(G_2)$, but I am not a hundred percent sure of this. It even seems like $\text{diam}(G_1\square G_2)= \text{diam}(G_1)+\text{diam}(G_2)$, but I am struggling to find a proof for this.