On the Spectral Determination of Complements of $T$-shape Trees

A graph $G$ is said to be \emph{determined by its spectrum} if every graph cospectral with $G$ is isomorphic to $G$. A \emph{T-shape tree} is defined as a tree containing exactly one vertex of maximum degree three. For any three positive integers ${\mathrm{\ell }}_1$,${\mathrm{\ell }}_2$ and ${\mathrm{\ell }}_3$ with ${\mathrm{\ell }}_1\le {\mathrm{\ell }}_2\le {\mathrm{\ell }}_3$, we denote by $T({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,{\mathrm{\ell }}_3)$ the unique $T$-shape tree such that deleting its degree-three vertex $v$ yields three disjoint paths $P_{{\mathrm{\ell }}_1}$, $P_{{\mathrm{\ell }}_2}$, and $P_{{\mathrm{\ell }}_3}$, i.e., $T({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,{\mathrm{\ell }}_3)-v=P_{{\mathrm{\ell }}_1}\cup P_{{\mathrm{\ell }}_2}\cup P_{{\mathrm{\ell }}_3}$, where $P_k$ stands for the path graph on $k$ vertices. In this paper, we establish a complete spectral characterization for the complements of $T$-shape trees, settling a long-standing conjecture posed by Wang and Xu (2006). Specifically, we prove that the complement of $T({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,{\mathrm{\ell }}_3)$ is spectrally determined if and only if $({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,{\mathrm{\ell }}_3)\mathrm{\notin }\{(\mathrm{\ell },\mathrm{\ell },2\mathrm{\ell }-2):\mathrm{\ell }\ge 2\}$. Moreover, all cospectral mates of the complement of the $T$-shape tree $T(\mathrm{\ell },\mathrm{\ell },2\mathrm{\ell }-2)$ are identified for every integer $\mathrm{\ell }\ge 2$.