# Some New Bounds for the Inverse Sum Indeg Energy of Graphs
###### Abstract  
Let \(G\) be a (molecular) graph with \(n\) vertices, and \(d_{i}\) be the degree of its \(i\)-th vertex. Then, the inverse sum indeg matrix of \(G\) is the \(n\times n\) matrix \(\mathcal{C}(G)\) with entries \(c_{ij}=\frac{d_{i}d_{j}}{d_{i}+d_{j}}\), if the \(i\)-th and the \(j\)-th vertices are adjacent and \(0\) otherwise. Let \(\mu_{1}\geq\mu_{2}\geq\ldots\geq\mu_{n}\) be the eigenvalues of \(\mathcal{C}\) arranged in order. The inverse sum indeg energy of \(G\), \(\varepsilon_{isi}(G)\) can be represented as \(\sum_{j=1}^{n}|\mu_{i}|\). In this paper, we establish several novel upper and lower sharp bounds on \(\mu_{1}\) and \(\varepsilon_{isi}(G)\) via some other graph parameters, and describe the structures of the extremal graphs. inverse sum indeg index; energy; ISI matrix; ISI energy 202211243390/axioms1105024310  
with the experimentally determined heats of formation and other measures of the thermodynamic stability of underlying conjugated compounds. Following in-depth research, it was found that this graph parameter can be successfully utilized in many fields, not only in chemistry [3; 7; 8]. In consideration of the successful development of the mathematical theory of graph energy, many extended graph energies have been gradually proposed based on the eigenvalues of other graph matrices, such as the (first) Zagreb matrix [9], the harmonic matrix [10], etc. ; see [3; 9; 11; 12; 13; 14; 15; 16; 17] and some more recent results to be found in [18; 19; 20]. In [21], [PERSON] et al. introduced the _Randic matrix_, \(R(G)=(r_{ij})_{n\times n}\), of a graph \(G\), where \(r_{ij}=\frac{1}{\sqrt{d_{i}d_{j}}}\) if \(v_{i}v_{j}\in E(G)\) and is 0 otherwise. Denote its eigenvalues by \(\rho_{1}\geq\rho_{2}\geq\ldots\geq\rho_{n}\). Then, in analogy to Equation (1), the _Randic energy_ is defined as  
\[RE=RE(G)=\sum_{i=1}^{n}\left|\rho_{i}\right|\,. \tag{2}\]  
The _extended adjacency matrix_ of graph \(G\), denoted by \(A_{ex}=A_{ex}(G)\), was put forward by [PERSON] et al. [22] and is defined so that its \((i,j)\) entry is equal to \(\frac{1}{2}(\frac{d_{i}}{d_{j}}+\frac{d_{j}}{d_{i}})\) if \(v_{i}v_{j}\in E(G)\) and is 0 otherwise. The _extended graph energy_ is defined as  
\[\varepsilon_{ex}=\varepsilon_{ex}(G)=\sum_{i=1}^{n}\left|\zeta_{i}\right|\,. \tag{3}\]  
where \(\zeta_{1}\geq\zeta_{2}\geq\ldots\geq\zeta_{n}\) are the ordered eigenvalues of \(A_{ex}\). In [12], [PERSON] et al. gave lower and upper bounds on the extended spectral radius \(\zeta_{1}\) and the extended energy \(\varepsilon_{ex}\) of graphs and the respective extremal graphs were characterized. Topological indices are of great importance to mathematical chemistry. A great deal of topological indices, such as the Randic index [23], atom-bond-connectivity index [24], sum-connectivity index [25], augmented Zagreb index [26], the eccentric-connectivity index [27], Zagreb indices [6; 28], the general eccentric-connectivity index [29], the general degree-eccentricity index [30], etc., were introduced to reveal the properties of organic compounds from different aspects. One of those numerical descriptors, the _inverse sum indeg index (ISI index_ for short) is an especially interesting vertex-degree-based topological index, which is defined as  
\[ISI(G)=\sum_{v_{i}v_{j}\in E(G)}\frac{d_{i}d_{j}}{d_{i}+d_{j}}.\]  
In 2010, [PERSON] and [PERSON] [31] proposed the ISI index, which can distinctively forecast the overall surface area of octane isomers. Similar to the Randic matrix and the extended adjacency matrix, [PERSON] et al. [32] and [PERSON] et al. [33] defined the _inverse sum indeg matrix_ (_ISI matrix_ for short) \(\mathcal{C}=\mathcal{C}(G)\) of a graph \(G\) as the matrix with entries:  
\[c_{ij}:=\left\{ \begin{array}{ll}\frac{d_{i}d_{j}}{d_{i}+d_{j}},&if\,\,v_{i} v_{j}\in E(G)\\ 0,&otherwise,\end{array} \right.\]  
respectively. Note that \(\mathcal{C}\) is a modification of the classical adjacency matrix involving the degrees of the vertices.