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arXiv · 2506.06934

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

Emily Barranca·Michael D. Barrus
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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 $P_2(1,n)$.

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Emily Barranca, Michael D. Barrus. 2025-06-07. A characterization of all graphs cospectral to the double star $P_2(1,n)$. https://arxiv.org/abs/2506.06934

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