A note on an infinite family of graphs with all different integral Laplacian eigenvalues

In this note, we give an infinite family of optimal graphs called $G^{+}(d,c)$. They are optimal in the sense that they have the maximum possible number of vertices for given a diameter $d$ and the so-called `outer multiset dimension’ $c$. We provide their spectra, which have the property that their Laplacian eigenvalues are all different and integral. Finally, we also obtained their eigenvectors.