MSL0 by varying individually $K_{0}$ (a), $E_{0}(\protect\rho _{0})$ (b), $m_{s,0}^{\ast }$ (c), and $\protect\rho _{0}$ (d).[]{data-label="XSigmaPara"}](XSigmaPara.eps)

To understand more clearly why the ISGMR energies decrease with $m_{s,0}^{\ast }$ observed in Fig. \[XEaPb208\], it is useful to note the fact that, with the standard Skyrme interaction, the $K_{0}$ and $m_{s,0}^{\ast }$ cannot be chosen independently if the Skyrme interaction parameter $\sigma $ in Eq. (\[V12Sky\]), $E_{0}(\rho _{0})$ and $\rho _{0}$ are fixed [@Boh79]. It should be stressed here that, instead of assuming a fixed value of $\sigma $ as in the usual parametrization and correlation analysis [@Cha97; @Col04], in the present work, the $\sigma $ parameter is determined by four macroscopic quantities, i.e., $K_{0}$, $E_{0}(\rho _{0})$, $m_{s,0}^{\ast }$ and $\rho _{0}$ as shown in Eq. (\[SkySigma\]), and thus $K_{0}$ and $m_{s,0}^{\ast }$ can be chosen independently. Neglecting the isospin dependence (assuming $N\approx Z)$, the nuclear breathing mode energy for medium and heavy nuclei can be approximated by [@Boh79] $$E_{\mathrm{ISGMR}}\approx \sqrt{\frac{\hbar ^{2}(K_{0}-63\sigma )}{m\left\langle r^{2}\right\rangle }}\text{ (}K_{0}\text{ in MeV).}
\label{EGMRBohigas}$$Eq. (\[EGMRBohigas\]) implies that the nuclear breathing mode energy can be closely related to both $K_{0}$ and $m_{s,0}^{\ast }$ if the parameter $\sigma $ is free and the values of $E_{0}(\rho _{0})$ and $\rho 