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The relationship between the tendencies towards rigidity (measured by shear modulus, <i>G</i>) and hardness (measured by Vickers hardness, <i>HV</i>) of early transition metal (ETM)-based refractory high-entropy alloys (RHEA) and bond parameters (i.e., valence electron concentration (<i>VEC</i>), enthalpy of mixing (Δ<i>H_mix</i>)) was investigated. These bond parameters, <i>VEC</i> and Δ<i>H_mix</i>, are available from composition and tabulated data, respectively. Based on our own data (9 samples) and those available from the literatures (47 + 27 samples), it seems that for ETM-based RHEAs the <i>G</i> and <i>HV</i> characteristics have a close correlation with the bonding parameters. The room temperature value of <i>G</i> and <i>HV</i> increases with the <i>VEC</i> and with the negative value of Δ<i>H_mix</i>. Corresponding equations were deduced for the first time through multiple linear regression analysis, in order to help design the mechanical properties of ETM refractory high-entropy alloys.