Constraining dark energy evolution in f(Q,C) gravity using observational datasets and geometric diagnostics

We investigate the cosmological implications of $f(Q,C)$ gravity with $f(Q,C)=\alpha Q+\beta C$, where Q is the non-metricity scalar and C encapsulates cosmological expansion terms. Three parameterizations of the EoS for dark energy, $\omega ={\omega }_0+{\omega }_{1}z$, $\omega ={\omega }_0+\frac{{\omega }_{1}z(1+z)}{1+z^2}$ and $\omega ={\omega }_0+\frac{{\omega }_1{z}^2}{1+z^2}$ are tested using the Hubble, Hubble+BAO, and Hubble+BAO+Pantheon datasets to constrain model parameters. The resulting Hubble and deceleration parameters reveal a transition from deceleration to acceleration, supporting current cosmic acceleration observations. Analysis of the energy density and pressure confirms positive energy density and a negative pressure for dark energy, potentially driving the late-time acceleration. We examine energy conditions, showing compliance with NEC, WEC and DEC, while SEC remains negative, supporting an accelerated expansion. Statefinder diagnostics suggest that two of the EoS parameterizations lead to Quintessence-like behavior with a time-varying dark energy component, while the third closely approaches ΛCDM showing slight deviations consistent with recent observations. Sound speed analysis demonstrates the physical stability of all parameterizations. Overall, our findings support $f(Q,C)$ gravity as a viable framework for describing diverse dark energy dynamics, providing insights into the Universe's accelerated expansion.