Spectral faux trees

TL;DR

This paper explores the existence and construction of spectral faux trees—graphs that are not trees but share the same spectral properties as trees for various matrices, revealing diverse possibilities depending on the matrix type.

Contribution

It provides new insights into spectral faux trees for different matrices, including constructions and limitations, especially for the adjacency and normalized adjacency matrices.

Findings

No spectral faux trees for Laplacian matrix.

Almost all trees are cospectral with a faux tree for adjacency matrix.

Spectral faux trees exist for normalized adjacency when n ≥ 4, with exponential growth in construction.

Abstract

A spectral faux tree with respect to a given matrix is a graph which is not a tree but is cospectral with a tree for the given matrix. We consider the existence of spectral faux trees for several matrices, with emphasis on constructions. For the Laplacian matrix, there are no spectral faux trees. For the adjacency matrix, almost all trees are cospectral with a faux tree. For the signless Laplacian matrix, spectral faux trees can only exist when the number of vertices is of the form $n=4k$. For the normalized adjacency, spectral faux trees exist when the number of vertices $n\ge 4$, and we give an explicit construction for a family whose size grows exponentially with $k$ for $n=\alpha k+1$ where $\alpha$ is fixed.