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Mathematics ~{\bf 8} (1)~(2023) $1-17$\\}  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{lemma}\label{thm:path}
Suppose that $T\not =P_{n}$ is a tree of order $n$. If $T^{\ast}$ is the tree of order $n$ obtained from $T\neq P_n$ applying Transformation A, then
%Let $T^{\ast}$ be the tree of order $n$ obtained from $T\neq P_n$ applying Transformation A. Then
\begin{equation}\label{a1}
\xi^{ad}(T)>\xi^{ad}(T^{\ast}).
\end{equation}
\end{lemma}
Let Lemma \eqref{a1}
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