A characterization of all graphs cospectral to the double star $P_2(1,n)$

TL;DR

This paper investigates the spectral properties of double star trees, characterizes graphs cospectral to them, provides constructions for certain cases, and proves that odd-n double stars are uniquely determined by their spectra.

Contribution

It offers a complete characterization of graphs cospectral to double star trees, including explicit constructions and a proof of spectral uniqueness for odd-n cases.

Findings

Constructed graphs cospectral to specific double stars.

Proved that P_2(1,n) is determined by its spectrum for odd n.

Identified all graphs sharing the spectrum with certain double stars.

Abstract

We examine the adjacency spectrum of trees with diameter three, also referred to as double stars. Using $P_2(a,b)$ to denote a double star with $ a$ and $b$ leaves at its respective endpoints, we discuss graphs which are cospectral to double stars for various parameters $a$ and $b$. In particular, we give constructions for graphs cospectral to $P_2(1,2k)$ for integers $k$. Lastly, we show that the double star $P_2(1,n)$ is determined by its spectrum when $n$ is odd. That is, if a graph $G$ cospectral to $P_2(1,n)$ for odd $n$, then $G$ is isomorphic to $P_2(1,n)$.