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This study is the first of a series of papers that provide a technique toanalyse the mixed-modes frequency spectra and characterise the structure ofstars on the subgiant and red-giant branches. We define seismic indicators,relevant of the stellar structure and study their evolution on a grid ofmodels. The proposed method, EGGMiMoSA, relies on the asymptotic description ofmixed modes, defines initial guesses for the parameters, and uses aLevenberg-Marquardt technique to adjust the mixed-modes pattern efficiently. Wefollow the evolution of the mixed-modes parameters along a grid of models fromthe subgiant phase to the RGB bump and extend past works. We show the impact ofthe mass and composition on their evolution. The evolution of the periodspacing $\Delta\pi_1$, pressure offset $\epsilon_p$, gravity offset$\epsilon_g$, and coupling factor $q$ as a function of $\Delta\nu$ is littleaffected by the chemical composition and it follows two different regimesdepending on the evolutionary stage. On the subgiant branch, the models displaya moderate core-envelope density contrast. The evolution of $\Delta \pi_1$,$\epsilon_p$, $\epsilon_g$, and $q$ thus significantly changes with the mass.Also, we demonstrate that, at fixed Z/X and with proper measurements of$\Delta\pi_1$ and $\Delta\nu$, we may unambiguously constrain the mass, radiusand age of a subgiant star. Conversely, on the red-giant branch, thecore-envelope density contrast becomes very large. Consequently, the evolutionof $\epsilon_p$, $\epsilon_g$ and $q$ as a function of $\Delta\nu$ becomesindependent of the mass. This is also true for $\Delta \pi_1$ in stars withmasses $\lesssim 1.8M_\odot$ because of core electron degeneracy. Thisdegeneracy is lifted for higher masses, again allowing for a precisemeasurement of the age. Overall, our computations qualitatively agree with pastobserved and theoretical studies.

Asteroseismology of evolved stars with EGGMiMoSA I. Theoretical mixed-mode patterns from the subgiant to the RGB phase