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

TL;DR

This paper introduces an infinite family of optimal graphs with maximum vertices for given diameter and outer multiset dimension, characterized by all distinct integral Laplacian eigenvalues and explicit eigenvectors.

Contribution

It constructs a new class of graphs with unique spectral properties, including all different integral Laplacian eigenvalues, and provides their spectra and eigenvectors.

Findings

Graphs have maximum vertices for given parameters

Laplacian eigenvalues are all different and integral

Eigenvectors are explicitly obtained

Abstract

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.