An undirected graph $G$ is determined by its $T$-gain spectrum (DTS) if every $T$-gain graph cospectral to $G$ is switching equivalent to $G$. We show that the complete graph $K_{n}$ and the graph $K_{n}-e$ obtained by deleting an edge from $K_{n}$ are DTS, the star $K_{1,n}$ is DTS if and only if $n\leq 2$, and an odd path $P_{2m+1}$ is not DTS if $m\geq 2$. We give an operation for constructing cospectral $T$-gain graphs and apply it to show that a tree of arbitrary order (at least $5$) is not DTS.

中文翻译：

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由它们的增益光谱确定的图形

无向图$G$由其决定$T$- 增益谱 (DTS) 如果每个$T$- 获得图谱到$G$切换相当于$G$. 我们证明了完整的图$K_{n}$和图表$K_{n}-e$通过删除一条边获得$K_{n}$是 DTS，明星$K_{1,n}$是 DTS 当且仅当$n\leq 2$, 和一条奇数路径$P_{2m+1}$不是 DTS 如果$m\geq 2$. 我们给出一个构造协谱的操作$T$- 获得图表并将其应用于显示任意顺序的树（至少$5$) 不是 DTS。