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

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 $P_2(1,n)$.