Laplacian $\{-1,0,1\}$- and $\{-1,1\}$-diagonalizable graphs

TL;DR

This paper characterizes graphs whose Laplacian matrices are diagonalizable by matrices with entries from specific sets, introducing balanced vectors and providing classifications for small graphs and various graph operations.

Contribution

It introduces the concept of balanced vectors to characterize diagonalizability of graphs by specific matrices and classifies small graphs with these properties.

Findings

Characterization of $ ext{-}1,0,1$-diagonalizable graphs using balanced vectors

Complete classification of small graphs up to nine vertices with these properties

Results on diagonalizability of graph operations like Cartesian product and join

Abstract

A graph is called "Laplacian integral" if the eigenvalues of its Laplacian matrix are all integers. We investigate the subset of these graphs whose Laplacian is furthermore diagonalized by a matrix with entries coming from a fixed set, in particular, the sets $\{-1,0,1\}$ or $\{-1,1\}$. Such graphs include as special cases the recently-investigated families of "Hadamard-diagonalizable" and "weakly Hadamard-diagonalizable" graphs. As a combinatorial tool to aid in our investigation, we introduce a family of vectors that we call "balanced", which generalizes totally balanced partitions, regular sequences, and complete partitions. We show that balanced vectors completely characterize which graph complements and complete multipartite graphs are $\{-1,0,1\}$-diagonalizable, and we furthermore prove results on diagonalizability of the Cartesian product, disjoint union, and join of graphs. Particular attention is paid to the $\{-1,0,1\}$- and $\{-1,1\}$-diagonalizability of the complete graphs and complete multipartite graphs. Finally, we provide a complete list of all simple, connected graphs on nine or fewer vertices that are $\{-1,0,1\}$- or $\{-1,1\}$-diagonalizable.