  In this work, we present a comprehensive exploration of the entanglement and
graph connectivity properties of graph states. We quantify the entanglement in
pseudo graph states using the entanglement distance, a recently introduced
measure of entanglement. Additionally, we propose a novel approach to probe the
underlying graph connectivity of genuine graph states, using quantum
correlators of Pauli matrices. Our findings also reveal interesting
implications for measurement processes, demonstrating the equivalence of
certain projective measurements. Finally, we emphasize the simplicity of data
analysis within this framework. This work contributes to a deeper understanding
of the entanglement and connectivity properties of graph states, offering
valuable insights for quantum information processing and quantum computing
applications. In this work, we do not resort to the celebrated stabilizer
formalism, which is the framework typically preferred for the study of this
type of state; on the contrary, our approach is solely based on the concepts of
expectation values, quantum correlations and projective measurement, which have
the advantage of being very intuitive and fundamental tools of quantum theory.
