# Some New Bounds for the Inverse Sum Indeg Energy of Graphs
## 4 Bounds for the ISI Energy of Graphs
Hence, inequality (25) becomes  
\[tr(\mathcal{C}^{2})=\sum_{i=1}^{n}\mu_{i}^{2}=\sum_{i=1}^{n}\gamma_{i}^{2}\.\]4  
Simplifying the above inequality, we obtain  
\[tr(\mathcal{C}^{2})=\sum_{i=1}^{n}\mu_{i}^{2}=\sum_{i=1}^{n}\gamma_{i}^{2}\.\]5  
Solving this quadratic inequality, we have  
\[tr(\mathcal{C}^{2})=\sum_{i=1}^{n}\mu_{i}^{2}=\sum_{i=1}^{n}\gamma_{i}^{2}\.\]6  
If \(G\cong\overline{K_{n}}\) or \(G\cong\frac{n}{2}K_{2}\), the equality in (26) is clearly attained. Conversely, suppose that equality holds in (26). Then, we have that \(\gamma_{1}=\gamma_{2}=\ldots=\gamma_{n}\). Therefore, \(G\cong\frac{n}{2}K_{2}\) or \(G\cong\overline{K_{n}}\). This completes the proof. In 1950, [PERSON], [PERSON] and [PERSON] [59] proved the following Gruss-type discrete inequality. **Lemma 21** ([59]). : _Let \(x_{1},x_{2},\ldots,x_{n}\) and \(y_{1},y_{2},\ldots,y_{n}\) be real numbers for which there exist real constants \(a,b,A\) and \(B\), so that for each \(i,i=1,2,\ldots,n\); \(a\leq x_{i}\leq A\) and \(b\leq y_{i}\leq B\). Then,_  
\[tr(\mathcal{C}^{2})=\sum_{i=1}^{n}\mu_{i}^{2}=\sum_{i=1}^{n}\gamma_{i}^{2}\.\]7_where \(\theta(n)=n[\frac{n}{2}](1-\frac{1}{n}[\frac{n}{2}])\), while \([x]\) denotes the integer part of a real number \(x\). Equality in Equation (27) holds if and only if \(x_{1}=x_{2}=\ldots=x_{n}\) and \(y_{1}=y_{2}=\ldots=y_{n}\)._  
Theorem 3.1: _For any graph \(G\) of order \(n\), the following inequality is valid_  
\[tr(\mathcal{C}^{2})=\sum_{i=1}^{n}\mu_{i}^{2}=\sum_{i=1}^{n}\gamma_{i}^{2}\.\]8  
_The equality holds if and only if \(G\cong\overline{K_{n}}\) or \(G\cong\frac{n}{2}K_{2}\)._  
Proof. : Applying inequality (27) by letting \(x_{i}=\gamma_{i}\), \(y_{i}=\gamma_{i}\), \(a=b=\gamma_{n}\) and \(A=B=\gamma_{1}\), we obtain  
\[tr(\mathcal{C}^{2})=\sum_{i=1}^{n}\mu_{i}^{2}=\sum_{i=1}^{n}\gamma_{i}^{2}\.\]9  
It implies that  
\[\epsilon_{isi}(G)\leq\frac{\Delta}{2}\sqrt{2 mn}\ . \tag{22}\]0  
From (23), we know that \(\epsilon_{isi}(G)\leq\sqrt{ntr(\mathcal{C}^{2})}\). Hence, we obtain  
\[\epsilon_{isi}(G)\leq\frac{\Delta}{2}\sqrt{2 mn}\ . \tag{22}\]1  
Since equality in (27) holds if and only if \(x_{1}=x_{2}=\ldots=x_{n}\) and \(y_{1}=y_{2}=\ldots=y_{n}\), equality in (29) holds if and only if \(\gamma_{1}=\gamma_{2}=\ldots=\gamma_{n}\). So, \(G\cong\frac{n}{2}K_{2}\) or \(G\cong\overline{K_{n}}\). Conversely, when \(G\cong\frac{n}{2}K_{2}\) or \(G\cong\overline{K_{n}}\) the equality is attained. This completes the proof. Theorem 3.2: _Let \(G\) be a graph of order \(n\) with minimum degree \(\delta\). Then,_  
\[\epsilon_{isi}(G)\leq\frac{\Delta}{2}\sqrt{2 mn}\ . \tag{22}\]2  
_The equality holds if and only if \(G\cong K_{n_{1},n_{2},\ldots,n_{t}}\), where \(|n_{1}|=|n_{2}|=\ldots=|n_{t}|\) and \(n=n_{1}+n_{2}+\ldots+n_{t}\)._  
Proof. : By Theorem 3.1, we have  
\[\epsilon_{isi}(G)\leq\frac{\Delta}{2}\sqrt{2 mn}\ . \tag{22}\]3  
where the second equality holds if and only if \(G\) is a regular graph. Therefore, \(\mu_{i}=\frac{\delta}{2}\lambda_{i}\) for any \(1\leq i\leq n\). The first equality holds if and only if \(G\) has only one positive eigenvalue in its adjacency spectrum. Therefore, from Lemma 3.1, the equality holds if and only if \(G\cong K_{n_{1},n_{2},\ldots,n_{t}}\), where \(|n_{1}|=|n_{2}|=\ldots=|n_{t}|\). This completes the proof. Given a graph \(G\), if all the eigenvalues in its adjacency spectrum are nonzero, then \(G\) is said to be _nonsingular_. Similarly, if all eigenvalues of the \(ISI\) matrix of \(G\) are nonzero, then \(G\) is called _ISI nonsingular_. Next, we give a lower bound on \(\epsilon_{isi}(G)\) for an \(ISI\) nonsingular connected graph \(G\). Lemma 3.3 ([60]). : _For any graph \(G\) of order \(n\) and size \(m\), we have_  
\[\epsilon_{isi}(G)\leq\frac{\Delta}{2}\sqrt{2 mn}\ . \tag{22}\]4  
_and equality is attained if and only if \(G\) is regular._  
**Theorem 10**. : _Let \(G\) be an 1S1 nonsingular connected graph of order \(n\) with \(\delta\geq 2\).