On inverse sum indeg energy of graphs

Abstract

Abstract For a simple graph with vertex set { v 1 , v 2 , … , v n } \left\{{v}_{1},{v}_{2},\ldots ,{v}_{n}\right\} and degree sequence d v i i = 1 , 2 , … , n {d}_{{v}_{i}}\hspace{0.33em}i=1,2,\ldots ,n , the inverse sum indeg matrix (ISI matrix) A ISI ( G ) = ( a i j ) {A}_{{\rm{ISI}}}\left(G)=\left({a}_{ij}) of G G is a square matrix of order n , n, where a i j = d v i d v j d v i + d v j , {a}_{ij}=\frac{{d}_{{v}_{i}}{d}_{{v}_{j}}}{{d}_{{v}_{i}}+{d}_{{v}_{j}}}, if v i {v}_{i} is adjacent to v j {v}_{j} and 0, otherwise. The multiset of eigenvalues τ 1 ≥ τ 2 ≥ ⋯ ≥ τ n {\tau }_{1}\ge {\tau }_{2}\hspace{0.33em}\ge \cdots \ge {\tau }_{n} of A ISI ( G ) {A}_{{\rm{ISI}}}\left(G) is known as the ISI spectrum of G G . The ISI energy of G G is the sum ∑ i = 1 n ∣ τ i ∣ \mathop{\sum }\limits_{i=1}^{n}| {\tau }_{i}| of the absolute ISI eigenvalues of G . G. In this article, we give some properties of the ISI eigenvalues of graphs. Also, we obtain the bounds of the ISI eigenvalues and characterize the extremal graphs. Furthermore, we construct pairs of ISI equienergetic graphs for each n ≥ 9 n\ge 9 .