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Furthermore, we prove that the Smith normal form of $W(D_n)$ is $$\\text{diag}[\\underbrace{1,1,\\ldots,1}_{\\lceil\\frac{n}{2}\\rceil},\\underbrace{2,2,\\ldots,2}_{\\lfloor\\frac{n}{2}\\rfloor-1},0]$$ when $4\\nmid n$. This confirms a recent conjecture in [W.Wang, F.Liu, W.Wang, Generalized spectral characterizations of almost controllable graphs, European J. 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We prove that $\\text{rank}\\,W(D_n)=n-2$ if $4\\mid n$ and $\\text{rank}\\,W(D_n)=n-1$ otherwise. Furthermore, we prove that the Smith normal form of $W(D_n)$ is $$\\text{diag}[\\underbrace{1,1,\\ldots,1}_{\\lceil\\frac{n}{2}\\rceil},\\underbrace{2,2,\\ldots,2}_{\\lfloor\\frac{n}{2}\\rfloor-1},0]$$ when $4\\nmid n$. This confirms a recent conjecture in [W.Wang, F.Liu, W.Wang, Generalized spectral characterizations of almost controllable graphs, European J. 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