## 2 Preliminary Results  
In this section, we give some preliminary results which will be utilized in the subsequent sections. The following lemma can be found in [7].  
**Lemma 1** ([7]).: _Suppose that \(G\) is a connected graph of order \(n\). We have_  
\[\partial(G)\geq\frac{2W(G)}{n},\]  
_where the equality holds if and only if \(G\) is transmission regular._  
By a similar way as used in the proof of ([32], Lemma 2), we can prove the following lemma.  
**Lemma 2**.: _A connected graph \(G\) admits two distinct generalized distance eigenvalues if and only if \(G\) is a complete graph._  
The proof of the following lemma is similar to that of ([4], Theorem 2.2) and is omitted here.  
**Lemma 3**.: _Let the transmission degree sequence of \(G\) be \(\{Tr_{1},Tr_{2},\ldots,Tr_{n}\}\). Then,_  
\[\partial(G)\geq\sqrt{\frac{\sum_{i=1}^{n}Tr_{i}^{2}}{n}},\]  
_where the equality holds if and only if \(G\) is transmission regular._  
Given two nonincreasing real sequences \((x)=(x_{1},x_{2},\ldots,x_{n})\) and \((y)=(y_{1},y_{2},\ldots,y_{n})\) of length \(n\), \((x)\) is said to be majorized by \((y)\) or \((y)\) majorizes \((x)\), denoted by \((x)\preceq(y)\) if  
\[\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}y_{i}\ \ \text{and}\ \ \ \sum_{i=1}^{k}x_{i}\leq\sum_{i=1}^{k}y_{i},\ \ \text{for all}\ \ k=1,2,\ldots,n-1.\]  
The relation \((x)\prec(y)\) means that \((x)\preceq(y)\) and \((x)\) is not the rearrangement of \((y)\).  
The following observation can be found in [23].  
**Lemma 4** ([23]).: _Let \((x)=(x_{1},x_{2},\ldots,x_{n})\) and \((y)=(y_{1},y_{2},\ldots,y_{n})\) be nonincreasing sequences of real numbers of length \(n\). If \((x)\preceq(y)\), then for any convex function \(\psi\), we have \(\sum_{i=1}^{n}\psi(x_{i})\leq\sum_{i=1}^{n}\psi(y_{i})\). Equality holds if and only if \(x_{i}=y_{i}\) for all \(i=1,2,\ldots,n\). In addition, when \((x)\prec(y)\) and \(\psi\) is strictly convex, \(\sum_{i=1}^{n}\psi(x_{i})<\sum_{i=1}^{n}\psi(y_{i})\) holds._  
The majorization relation between the spectrum and diagonal elements of Hermitian matrices perfectly links the spectrum of a generalized distance matrix \(D_{\alpha}(G)\) with majorization. The relation given below immediately follows.  
**Lemma 5**.: _Assume that \(G\) is a connected graph of order \(n\) admitting generalized distance eigenvalues \(\partial_{1},\partial_{2},\ldots,\partial_{n}\) and transmission degrees \(Tr_{1},Tr_{2},\ldots,Tr_{n}\). Then,_  
\[(\alpha Tr_{1},\alpha Tr_{2},\ldots,\alpha Tr_{n})\preceq(\partial_{1}, \partial_{2},\ldots,\partial_{n}).\]  
**Theorem 1**.: _Assume that \(G\) is any connected graph of order \(n\geq 2\) with transmission degree sequence \(\{Tr_{1},Tr_{2},\ldots,Tr_{n}\}\)._  
(i) _If_ \(k<0\) _or_ \(k>1\)_, then_ \(U_{k}\geq\sum\limits_{i=1}^{n}\left(\alpha Tr_{i}-\frac{2\alpha W(G)}{n} \right)^{k}\)_;_  
(ii) _If_ \(0<k<1\)_, then_ \(U_{k}\leq\sum\limits_{i=1}^{n}\left(\alpha Tr_{i}-\frac{2\alpha W(G)}{n} \right)^{k}\)_._  
_Equality occurs in both parts, if and only if_ \(\partial_{i}=\alpha Tr_{i}\)_, for_ \(i=1,2,\ldots,n\)_._  
Proof.: (i) For \(x>0\), it follows that the function \(f(x)=x^{k}\) is a convex function if \(k<0\) or \(k>1\). Let \((X)=\left(\alpha Tr_{1}-\frac{2\alpha W(G)}{n},\alpha Tr_{2}-\frac{2\alpha W( G)}{n},\ldots,\alpha Tr_{n}-\frac{2\alpha W(G)}{n}\right)\) and \((Y)=\left(\partial_{1}-\frac{2\alpha W(G)}{n},\partial_{2}-\frac{2\alpha W(G) }{n},\ldots,\partial_{n}-\frac{2\alpha W(G)}{n}\right)\). By Lemma 5, we have \((X)\preceq(Y)\). It then follows by Lemma 4 that  
\[U_{k}=\sum\limits_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{ k}\geq\sum\limits_{i=1}^{n}\left(\alpha Tr_{i}-\frac{2\alpha W(G)}{n}\right)^{k}. \tag{3}\]  
Therefore, by Lemma 4, equality occurs in the inequality (3) if and only if \((X)=(Y)\). That is, if and only if \(\partial_{i}=\alpha Tr_{i}\), for all \(i=1,2,\ldots,n\).  
(ii) For \(x>0\), it follows that the function \(f(x)=-x^{k}\) is a convex function if \(0<k<1\). Therefore, proceeding similarly as in part (i), we arrive at part (ii). \(\square\)