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The transport coefficients of strongly interacting matter are currentlysubject of intense theoretical and phenomenological studies due to theirrelevance for the characterization of the quark-gluon plasma produced in ultrarelativistic heavy-ion collisions (uRHIC). We discuss the connection betweenthe shear viscosity to entropy density ratio, $\eta/s$, and the electricconductivity, $\sigma_{el}$. Once the relaxation time is tuned to have aminimum value of $\eta/s=1/4\pi$ near the critical temperature $T_c$, onesimultaneously predicts $\sigma_{el}/T$ very close to recent lQCD data. Moregenerally, we discuss why the ratio of $(\eta/s)/(\sigma_{el}/T)$ supplies ameasure of the quark to gluon scattering rates whose knowledge would allow tosignificantly advance in the understanding of the QGP phase. We also predictthat $(\eta/s)/(\sigma_{el}/T)$, independently on the running coupling$\alpha_s(T)$, should increase up to about $\sim 20$ for $T \rightarrow T_c$,while it goes down to a nearly flat behavior around $\simeq 4$ for $T \geq 4\,T_c$. Therefore we in general predict a stronger T dependence of$\sigma_{el}/T$ with respect to $\eta/s$ that in a quasiparticle approach isconstrained by lQCD thermodynamics. A conformal theory, instead, predicts asimilar T dependence of $\eta/s$ and $\sigma_{el}/T$.

Shear viscosity $\eta$ to electric conductivity $\sigma_{el}$ ratio for the Quark-Gluon Plasma