Inverse Sum Indeg Energy of Graphs

Suppose <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-vertex simple graph with vertex set <inline-formula> <tex-math notation="LaTeX">$\{v_{1}, {\dots },v_{n}\}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$d_{i}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$i=1, {\dots },n$ </tex-math></inline-formula>, is the degree of vertex <inline-formula> <tex-math notation="LaTeX">$v_{i}$ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. The ISI matrix <inline-formula> <tex-math notation="LaTeX">$S(G)= [s_{ij}]_{n\times n}$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is defined by <inline-formula> <tex-math notation="LaTeX">$s_{ij}= \frac {d_{i} d_{j}}{d_{i}+d_{j}}$ </tex-math></inline-formula> if the vertices <inline-formula> <tex-math notation="LaTeX">$v_{i}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$v_{j}$ </tex-math></inline-formula> are adjacent and <inline-formula> <tex-math notation="LaTeX">$s_{ij}=0$ </tex-math></inline-formula> otherwise. The <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>-eigenvalues of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> are the eigenvalues of its ISI matrix <inline-formula> <tex-math notation="LaTeX">$S(G)$ </tex-math></inline-formula>. Recently, the notion of inverse sum indeg (henceforth, ISI) energy of graphs is introduced and is defined by <inline-formula> <tex-math notation="LaTeX">$\sum \limits _{i=1}^{n}|\tau _{i}|$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\tau _{i}$ </tex-math></inline-formula> are the <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>-eigenvalues. We give ISI energy formula of some graph classes. We also obtain some bounds for ISI energy of graphs. In the end, we give some noncospectral equienergetic graphs with respect to inverse sum indeg energy.

Abstract licence: CC BY-NC-ND

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