On integer matrices with integer eigenvalues and Laplacian integral graphs

Abstract

A matrix A is said to be an integer matrix if all its entries are integers. In this article, we characterize the nonsingular integer matrices with integer eigenvalues in an expressible form of their corresponding inverse matrices. It is proved that a nonsingular integer matrix A has integer eigenvalues if and only if A−1 can be written as the sum of n rank-one matrices that meet certain requirements. A method for constructing integer matrices with integer eigenvalues using the Hadamard product is also provided. Let S represent an n-tuple of nonnegative integers. If there is an n×n integer matrix A whose spectrum (the collection of eigenvalues) is S, we say that S is realisable by an integer matrix. In Fallat et al. (2005) [7], the authors posed a conjecture that “there is no simple graph onn≥2vertices whose Laplacian spectrum is given by(0,1,…,n−1).” We provide a characterization of threshold graphs using the spectra of quotient matrices of its G-join graphs. As a consequence, we prove that given any n−1 positive integers λ2,…,λn such that λ2≤⋯≤λn, the n-tuple (0,λ2,…,λn) is realizable by the Laplacian matrix of a multidigraph. In particular, we show that (0,1,…,n−1) can be realizable by the Laplacian matrix of a multidigraph.