A simplified graph parameter and its relationship to the modified Randić index

Abstract

Recently, Lin introduced a new graph parameter $\xi$ defined by nine properties and established the inequality $\xi \left(G\right)\le 2H\left(G\right)$, where $H\left(G\right)$ is the harmonic index of $G$. In this note, we simplify the framework by reducing the required properties to five, defining a new parameter $\zeta$. We prove $\zeta \left(G\right)\le 2R^{\prime }\left(G\right)$, where $R^{\prime }\left(G\right)$ is the modified Randić index, and characterize the equality case. It is known that $R^{\prime }\left(G\right)\le H\left(G\right)\le R\left(G\right)$. Then $\zeta \left(G\right)\le 2R^{\prime }\left(G\right)\le 2H\left(G\right)\le 2R\left(G\right)$. Since $\zeta$ needs fewer properties than $\xi$, it has a high possibility that many known parameters satisfy the demands of $\zeta$.