On the Spectral Determination of Complements of \(T\)-shape Trees

Abstract

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 \(\ell_1\),\(\ell_2\) and \(\ell_3\) with \( \ell_1\leq \ell_2\leq \ell_3\), we denote by \(T(\ell_1,\ell_2,\ell_3)\) the unique \(T\)-shape tree such that deleting its degree-three vertex \(v\) yields three disjoint paths \(P_{\ell_1}\), \(P_{\ell_2}\), and \(P_{\ell_3}\), i.e., \(T(\ell_1,\ell_2,\ell_3)-v = P_{\ell_1}\cup P_{\ell_2}\cup P_{\ell_3}\), 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 \(T(\ell_1,\ell_2,\ell_3)\) is spectrally determined if and only if \((\ell_1,\ell_2,\ell_3) \notin \{(\ell,\ell,2\ell-2):\ell\ge2\}\). Moreover, all cospectral mates of the complement of the \(T\)-shape tree \(T(\ell,\ell,2\ell-2)\) are identified for every integer \(\ell\geq 2\).