Constructions of graphs and trees with partially prescribed spectrum

TL;DR

This paper presents methods to construct graphs and trees with specific spectral properties, demonstrating that any graph's characteristic polynomial can divide that of a suitably constructed tree, based on recent algebraic integer results.

Contribution

It introduces a novel construction approach for graphs and trees with partially prescribed spectra, leveraging Salez's result on algebraic integers as eigenvalues of trees.

Findings

Any graph's characteristic polynomial divides that of a constructed tree.

Constructed trees can realize prescribed spectral properties.

The approach extends spectral graph theory with new construction techniques.

Abstract

It is shown how a connected graph and a tree with partially prescribed spectrum can be constructed. These constructions are based on a recent result of Salez that every totally real algebraic integer is an eigenvalue of a tree. Our result implies that for any (not necessarily connected) graph $G$, there is a tree $T$ such that the characteristic polynomial $P(G,x)$ of $G$ can divide the characteristic polynomial $P(T,x)$ of $T$, i.e., $P(G,x)$ is a divisor of $P(T,x)$.