# Some New Bounds for the Inverse Sum Indeg Energy of Graphs
## 2 Preliminaries  
In this part, we give some lemmas which will come in handy in later parts. Lemma 1 ([32]): _For any connected graph \(G\) and every edge \(v_{i}v_{j}\in E(G)\), we have_  
\[\frac{\delta}{2}\leq\frac{d_{i}d_{j}}{d_{i}+d_{j}}\leq\frac{\Delta}{2}. \tag{5}\]  
_the equality in left and right hands are both attained iff \(G\) is regular._  
Lemma 2 ((Cauchy-Schwarz inequality) [47]): _Let \(W\) and \(Z\) be two \(n\)-dimension vectors with elements \(w_{i}\in\mathbb{R}\) and \(z_{i}\in\mathbb{R}(1\leq i\leq n)\), respectively. Then_  
\[\left(\sum_{i=1}^{n}w_{i}z_{i}\right)^{2}\leq\sum_{i=1}^{n}w_{i}^{2}\ \sum_{i=1}^{n}z_{i}^{2}\, \tag{6}\]  
_with equality iff there is a real number \(d\) satisfying that \(w_{j}=dz_{j}\ (1\leq j\leq n)\)._  
Lemma 3 ((Chebyshev's inequality) [48]): _For two sequences of \(w_{i}\in\mathbb{R}\) and \(z_{i}\in\mathbb{R}(1\leq i\leq n)\), such that \(w_{1}\leq w_{2}\leq\ldots\leq w_{n}\) and \(z_{1}\leq z_{2}\leq\ldots\leq z_{n}\), we have_  
\[\left(\sum_{i=1}^{n}w_{i}\right)\left(\sum_{i=1}^{n}z_{i}\right)\leq n\sum_{i= 1}^{n}w_{i}z_{i}\, \tag{7}\]  
_the equality is obtained iff \(w_{1}=w_{2}=\ldots=w_{n}\) or \(z_{1}=z_{2}=\ldots=z_{n}\)._  
**Lemma 4** ([49]). : _Let \(w_{i}(1\leq i\leq n)\) be real numbers satisfying \(w_{n}\leq w_{n-1}\leq\ldots\leq w_{1}\). Then_  
\[\frac{\sum_{i=1}^{n}w_{i}}{n}+\sqrt{\frac{1}{n(n-1)}\sum_{i=1}^{n} \left(w_{i}-\frac{\sum_{i=1}^{n}w_{i}}{n}\right)^{2}}\leq w_{1}. \tag{8}\]  
**Lemma 5** ([50]). : _Let \(G\) be a connected \((n,m,\delta,\Delta)\)-graph, for any \(v_{i}v_{j}\in E(G)\), we have_  
\[\frac{2\sqrt{\delta\Delta}}{\Delta+\delta}\leq\frac{2\sqrt{d_{i}d_{j}}}{d_{i} +d_{j}}\leq 1.\]  
_The equality in the left hand is attained iff \(G\) is \((\Delta,\delta)\)-semiregular or regular; the equality in the right hand is achieved iff \(G\) is regular._  
Recall that for any \(n\)-order square matrices \(M=(s_{jk})\) and \(N=(t_{jk})\), if \(s_{jk}\geq t_{jk}\) holds for any \(j,k\), then \(M\geq N\). **Lemma 6** ([51]). : _If \(M\geq N\) for any two symmetric, non-negative \(n\)-order square matrices \(M\) and \(N\), then \(\rho_{1}(M)\geq\rho_{1}(N)\), where \(\rho_{1}\) is the maximum eigenvalue._  
**Lemma 7** ((Interlacing Lemma) [52]). : _If \(M\) is a symmetric \(n\)-order square matrix, and \(M_{j}\) is the \(j\times j\) submatrix of \(M\), then, for any integer \(k\), \(1\leq k\leq j\),_  
\[\rho_{n-k+j}(M)\leq\rho_{i}(M_{j})\leq\rho_{k}(M), \tag{9}\]  
_where \(\rho_{k}(M_{j})\), \(\rho_{k}(M)\) are the \(k\)-th greatest eigenvalue of \(M\) and \(M_{j}\), respectively._  
**Lemma 8** ([53]). : _If \(G\) is an \(n\)-vertex graph having degree collection \(d_{i}(1\leq j\leq n)\), then_  
\[\lambda_{1}\geq\sqrt{\frac{\sum_{i=1}^{n}d_{i}^{2}}{n}}\, \tag{10}\]  
_the equality is achieved iff \(G\) is regular or semiregular._  
**Lemma 9** ((Rayleigh-Ritz) [54]). : _Let \(M\) be a real symmetric \(n\)-order square matrix having eigenvalues \(\rho_{1}\geq\rho_{2}\geq\ldots\geq\rho_{n}\), then, for a nonzero vector \(\mathbf{y}\),_  
\[\rho_{1}\geq\frac{\mathbf{y}^{T}M\mathbf{y}}{\mathbf{y}^{T}\mathbf{y}}\, \tag{11}\]  
_the equality holds iff \(\mathbf{y}\) is an eigenvector for \(\rho_{1}\) of \(M\)._  
**Lemma 10** ([55]). : _For any \(n\)-vertex graph \(G\), \(\varepsilon_{isi}(G)=0\) iff \(G\cong\overline{K_{n}}\)._  
**Lemma 11** ([56]). : _For any \((n,m,\delta,\Delta)\)-graph \(G\) such that \(\delta\geq 1\),_  
\[\lambda_{1}\leq\sqrt{2m-\delta(n-1)+(\delta-1)\Delta}\, \tag{12}\]  
_the equality is acquired iff \(G\) is regular, or the union of \(K_{1,n-3}\) and \(K_{2}\), or the union of a regular graph possessing smaller vertex degree and a complete graph._  
**Lemma 12**. : _Let \(G\) be an \(n\)-vertex connected graph. Then \(\mu_{1}>\mu_{2}\)._  
Proof. : Suppose the result is false. Let \(\mathbf{u}\) and \(\mathbf{v}\) be eigenvectors corresponding to \(\mu_{1}\) and \(\mu_{2}\), respectively. Note that the fact that \(G\) is connected.