On inverse sum indeg energy of graphs

Abstract

For a simple graph with vertex set {v1,v2,…,vn}\left\{{v}_{1},{v}_{2},\ldots ,{v}_{n}\right\} and degree sequence dvii=1,2,…,n{d}_{{v}_{i}}\hspace{0.33em}i=1,2,\ldots ,n, the inverse sum indeg matrix (ISI matrix) AISI(G)=(aij){A}_{{\rm{ISI}}}\left(G)=\left({a}_{ij}) of GG is a square matrix of order n,n, where aij=dvidvjdvi+dvj,{a}_{ij}=\frac{{d}_{{v}_{i}}{d}_{{v}_{j}}}{{d}_{{v}_{i}}+{d}_{{v}_{j}}}, if vi{v}_{i} is adjacent to vj{v}_{j} and 0, otherwise. The multiset of eigenvalues τ1≥τ2≥⋯≥τn{\tau }_{1}\ge {\tau }_{2}\hspace{0.33em}\ge \cdots \ge {\tau }_{n} of AISI(G){A}_{{\rm{ISI}}}\left(G) is known as the ISI spectrum of GG. The ISI energy of GG is the sum ∑i=1n∣τi∣\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≥9n\ge 9.

Keywords

• topological indices
• adjacency matrix
• inverse sum indeg matrix
• energy
• 05c50
• 05c09
• 05c92
• 15a18