# Some New Bounds for the Inverse Sum Indeg Energy of Graphs
## 2 Preliminaries
If \(\mu_{1}=\mu_{2}\), by Perron-Frobeniustheorem, all elements of \(\mathbf{u}\) are positive. Since \(\mu_{1}=\mu_{2}\), any linear combination of \(\mathbf{u}\) and \(\mathbf{v}\) has to be an eigenvector for \(\mu_{1}\). In this way, one element of the vector can be adjusted to zero easily, a contradiction. Lemma 13: _Let \(G\) be an \(n\)-vertex graph.Then, \(|\mu_{1}|=|\mu_{2}|=\ldots=|\mu_{n}|\) if and only if \(G\cong\overline{K_{n}}\) or \(G\cong\frac{n}{2}K_{2}\)._  
Proof. : For convenience, we use \(I\) to stand for the collection of isolated vertices of \(G\). First, we assume \(|\mu_{1}|=|\mu_{2}|=\ldots=|\mu_{n}|\). Let \(k=|I|\). If \(k\geq 1\), then \(\mu_{1}=\mu_{2}=\ldots=\mu_{n}=0\), i.e., \(G\cong\overline{K_{n}}\). Otherwise, \(k=0\). If \(\Delta=1\), then \(d_{i}=1(1\leq i\leq n)\) and therefore \(G\cong\frac{n}{2}K_{2}\). Otherwise, \(\Delta\geq 2\), then \(G\) includes a connected component \(H\) such that \(|V(H)|\geq 3\). If \(H\cong K_{p}\)\((p\geq 3)\), then \(\mu_{1}(H)=\frac{(p-1)^{2}}{2}>\frac{p-1}{2}=\mu_{2}(H)\), a contradiction. Otherwise, \(H\
ot\cong K_{p}\)\((p\geq 3)\), then \(diam(H)\geq 2\). We suppose that \(G\) contain an induced shortest path \(P_{m}\), \(m\geq 2\). Let \(B\) be the principal submatrix of \(\mathcal{C}\) indexed by the vertices of \(P_{m}\) and then by the interlacing theorem 7 we obtain \(\mu_{2}(H)\geq\mu_{2}(B)\geq 0\). Moreover, by Lemma 12, we know that \(\mu_{1}(H)>\mu_{2}(H)\), a contradiction. On the contrary, when \(G\cong\overline{K_{n}}\) or \(G\cong\frac{n}{2}K_{2}\), we have \(|\mu_{1}|=|\mu_{2}|=\ldots=|\mu_{n}|\). Lemma 14 ([32]): _For any \(n\)-vertex graph \(G\) having vertices \(v_{j}(1\leq j\leq n)\), we have_  
_(1)_ \(tr(\mathcal{C})=0\) _;_  
_(2)_  
\[tr(\mathcal{C}^{2})=2\sum_{v_{j}\in\mathcal{E}(G)}\frac{d_{i}^{2}d_{j}^{2}}{( d_{i}+d_{j})^{2}}\;;\]  
_(3)_  
\[\left(\sum_{i=1}^{n}w_{i}z_{i}\right)^{2}\leq\sum_{i=1}^{n}w_{i}^{2}\ \sum_{i=1}^{n}z_{i}^{2}\, \tag{6}\]0  
\[+\sum_{ \begin{subarray}{c}v_{j}v_{j}\in\mathcal{V}(G)\\ v_{j}\
eq v_{j}\end{subarray} }d_{i}^{2}d_{j}^{2}\left(\sum_{ \begin{subarray}{ c}v_{k}\in\mathcal{V}(G)\\ v_{k}\sim v_{j}\end{subarray} }\frac{d_{k}^{2}}{(d_{i}+d_{k})(d_{k}+d_{j})} \right)^{2}.\]  
Lemma 15 ([9]): _For integers \(z_{i}\geq 0(1\leq i\leq n)\) and \(s\geq 2\), we get_  
\[\sum_{i=1}^{n}(z_{i})^{s}\leq\left(\sum_{i=1}^{n}z_{i}^{2}\right)^{\frac{s}{2}}, \tag{13}\]  
_the equality is obtained iff \(z_{1}=z_{2}=\ldots=z_{n}\)._  
Lemma 16 ([57]): _Let \(G\) be an \(n\)-vertex connected graph. Then there exists just one positive eigenvalue in \(SpA(G)\) iff \(G\) is isomorphic to \(K_{1,l_{2},\ldots,l_{s}}\), \(n=l_{1}+l_{2}+\ldots+l_{s}\)._  
Lemma 17 ([9]): _Let \(y_{i}>0(1\leq i\leq p)\) in \(\mathbb{R}\). Then_  
\[\frac{p}{\frac{1}{y_{1}}+\frac{1}{y_{2}}+\ldots+\frac{1}{y_{p}}}\,\leq\, \bar{\psi}\gamma y_{1}y_{2}\ldots y_{p}\,. \tag{14}\]  
The _nullity_\(n_{0}(G)\) of a graph \(G\) is the multiplicity of eigenvalue \(0\) in its adjacency spectrum. Lemma 18 ([48]): _For any \(n(n\geq 2)\)-vertex graph \(G\), \(n_{0}(G)=n-2\) iff \(G\cong K_{s,t}\cup(n-s-t)K_{1}\), \(s+t\leq n\)._  
**Lemma 19** ([55]). : _For any graph \(G\) with components \(G_{i},(1\leq i\leq k)\), we have_  
\[\varepsilon_{isi}(G)=\sum_{i=1}^{k}\varepsilon_{isi}(G_{i})\;.\]