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Low amplitude is the defining characteristic of solar-like oscillations. Thespace projects $Kepler$ and $CoRoT$ give us a great opportunity to successfullydetect such oscillations in numerous targets. Achievements of asteroseismologydepend on new discoveries of connections between the oscillation frequenciesand stellar properties. In the previous studies, the frequency of the maximumamplitude and the large separation between frequencies were used for thispurpose. In the present study, we confirm that the large separation between thefrequencies has two minima at two different frequency values. These are thesignatures of the He {\small II} ionization zone, and as such have very strongdiagnostic potential. We relate these minima to fundamental stellar propertiessuch as mass, radius, luminosity, age and mass of convective zone. For mass,the relation is simply based on the ratio of the frequency of minimum $\Delta\nu$ to the frequency of maximum amplitude. These frequency comparisons can bevery precisely computed, and thus the mass and radius of a solar-likeoscillating star can be determined to high precision. We also develop a newasteroseismic diagram which predicts structural and evolutionary properties ofstars with such data. We derive expressions for mass, radius, effectivetemperature, luminosity and age in terms of purely asteroseismic quantities.For solar-like oscillating stars, we now will have five very importantasteroseismic tools (two frequencies of minimum $\Delta \nu$, the frequency ofmaximum amplitude, and the large and small separations between the oscillationfrequencies) to decipher properties of stellar interior astrophysics.

Fundamental properties of solar-like oscillating stars from frequencies of minimum $\Delta \nu$: I. Model computations for solar composition