  The interplay between the quantum state space and a specific set of
measurements can be effectively captured by examining the set of jointly
attainable expectation values. This set is commonly referred to as the (convex)
joint numerical range. In this work, we explore geometric properties of this
construct for measurements represented by tensor products of Pauli observables,
also known as Pauli strings. The structure of pairwise commutation and
anticommutation relations among a set of Pauli strings determines a graph $G$,
sometimes also called the frustration graph. We investigate the connection
between the parameters of this graph and the structure of minimal ellipsoids
encompassing the joint numerical range. Such an outer approximation can be very
practical since ellipsoids can be handled analytically even in high dimensions.
  We find counterexamples to a conjecture from [C. de Gois, K. Hansenne and O.
G\"uhne, arXiv:2207.02197], and answer an open question in [M. B. Hastings and
R. O'Donnell, Proc. STOC 2022, pp. 776-789], which implies a new graph
parameter that we call $\beta(G)$. Besides, we develop this approach in
different directions, such as comparison with graph-theoretic approaches in
other fields, applications in quantum information theory, numerical methods,
properties of the new graph parameter, etc. Our approach suggests many open
questions that we discuss briefly at the end.
