T形树补图的谱确定
On the Spectral Determination of Complements of \(T\)-shape Trees
AI总结：
本文解决了Wang和Xu2006年的猜想，确定了T形树补图的谱判定条件及同谱图，证明其补图仅当(ℓ₁,ℓ₂,ℓ₃)不属于特定集合时为谱确定。
AI中文摘要：
若图G的所有与它谱相同的图都与G同构，则称G是由其谱确定的。T形树是指恰好含一个最大度为3的顶点的树。对任意满足ℓ₁≤ℓ₂≤ℓ₃的三个正整数，记T(ℓ₁,ℓ₂,ℓ₃)为唯一的T形树，其删除度为3的顶点v后得到三条不交路径P_{ℓ₁}、P_{ℓ₂}、P_{ℓ₃}，即T(ℓ₁,ℓ₂,ℓ₃)-v = P_{ℓ₁}∪P_{ℓ₂}∪P_{ℓ₃}，其中P_k表示含k个顶点的路径图。本文对T形树的补图建立了完整的谱刻画，解决了Wang和Xu于2006年提出的长期猜想。具体而言，证明了T(ℓ₁,ℓ₂,ℓ₃)的补图是谱确定的当且仅当(ℓ₁,ℓ₂,ℓ₃) ∉ {(ℓ,ℓ,2ℓ-2):ℓ≥2}，且对每个整数ℓ≥2，确定了T形树T(ℓ,ℓ,2ℓ-2)的补图的所有同谱图。
英文摘要：
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 \(\ell\geq 2\).
