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We conduct a detailed analysis of quasinormal $f-$mode frequencies in neutronstars (NS), within the linearized General Relativistic formalism. From Bayesianinference, we derived approximately 9000 nuclear Equations of State (EOS)subject to various constraints including nuclear saturation properties, thepure neutron matter EOS constraint obtained within $\chi$EFT, and pQCD atdensities relevant to NS cores. The composition and oscillatory dynamics of NSare then investigated using this set. The EOS are transformed into a spectralrepresentation, aiding in the efficient computation of NS properties. Themedian frequency values of the $f-$mode for NS with masses ranging from1.4$M_\odot$ to 2.0$M_{\odot}$ lie between 1.80 and 2.20 kHz for our entire EOSset. Our findings do not reveal a strong correlation between $f-$modefrequencies and individual nuclear saturation properties of the EOS. Thissuggests the need for more complex methods to unravel multiple-parameterrelationships. We noticed a strong relationship between the radii and $f-$modefrequencies for different NS masses. Using this correlation along with NICERobservations of PSR J0740+6620 and PSR 0030+0451, we obtained constraints thathave minimal overlap in the radius domain and differ in the frequency domainfrom our entire nucleonic EOS set. This indicates that there may be a need toconsider additional exotic particles or maybe a deconfined quark phase in theEOS relevant to the NS core. We argue that future observations of the radius or$f-$mode frequency for more than one NS mass, particularly at the extremes, arelikely to settle the issue by either ruling out only nucleonic EOS or providingdefinitive evidence in its favour.

Analysis of Neutron Star $f-$mode Oscillations in General Relativity with Spectral Representation of Nuclear Equations of State