in Eq. (\[KA1\]), one can express $K_{A}(N,Z)$ as $$\begin{aligned}
K_{A}(N,Z) &=&K_{0}+K_{\mathrm{surf}}A^{-1/3}+K_{\tau }\left( \frac{N-Z}{A}\right) ^{2}  \notag \\
&&+K_{\mathrm{Coul}}\frac{Z^{2}}{A^{4/3}},  \label{KA2}\end{aligned}$$where $K_{0}$, $K_{\mathrm{surf}}$, $K_{\tau }$, and $K_{\mathrm{coul}}$ represent the volume, surface, symmetry, and Coulomb terms, respectively. The $K_{\tau }$ parameter is usually thought to be equivalent to the isospin dependent part, i.e., the $K_{\mathrm{sat,2}}$ parameter, of the isobaric incompressibility coefficient of ANM (incompressibility evaluated at the saturation density of ANM) defined as$$K_{\mathrm{sat}}(\delta )=K_{0}+K_{\mathrm{sat,2}}\delta ^{2}+O(\delta ^{4}).
\label{Ksat}$$We would like to point out that the $K_{\mathrm{sat,2}}$ parameter is a theoretically well-defined physical property of ANM [@Pie09; @Che09] while the value of the $K_{\tau }$ parameter may depend on the details of the truncation scheme in Eq. (\[KA1\]) [@Bla81; @Sha88; @Pea91; @Shl93; @Pea10]. Here, we assume $K_{\mathrm{sat,2}}$ has similar influences on $K_{A}(N,Z)$ as $K_{\tau }$ and thus $K_{\mathrm{sat,2}}$ will affect the $E_{\mathrm{ISGMR}}$ through Eq. (\[EGMRKa\]), and then we can analyze the $L$ and $m_{s,0}^{\ast }$ dependences of $E_{\mathrm{ISGMR}}$ from the correlations of $K_{\mathrm{sat,2}}$ parameter with $L$ and $m_{s,0}^{\ast }$.

![The $K_{\mathrm{sat,2}}$ parameter obtained from SHF with MSL0 by varying individually $L$ (a) and $m_{s,0}^{\ast }$ (b).[]{data-label="XKsat2"}](XKsat2.eps)

The effects of the density dependence of the symmetry energy on the ISGMR centroid energy $E_{ave}$ of $^{208}$Pb has been 