  Experimental measurements of the residual resistivity $\rho(x)$ of the binary
alloy system Fe$_{1-x}$Cr$_x$ have shown an anomalous concentration dependence
which deviates significantly from Nordheim's rule. In the low ($x < 10%$) Cr
concentration regime the resistivity has been found to increase linearly with
$x$ until $\approx$ 10% Cr where the resistivity reaches a plateau persisting
to $\approx$ 20% Cr. In this paper we present $ab$-$initio$ calculations of
$\rho(x)$ which explain this anomalous behavior and which are based on the
Korringa-Kohn-Rostoker (KKR) method in conjunction with the Kubo-Greenwood
formalism. Furthermore we are able to show that the effects of short-range
ordering or clustering have little effect via our use of the nonlocal
coherent-potential approximation (NL-CPA). For the interpretation of the
results we study the alloys' electronic structure by calculating the Bloch
spectral function particularly in the vicinity of the Fermi energy. From the
analysis of our results we infer that a similar behavior of the resistivity
should also be obtained for iron-rich Fe$_{1-x}$V$_x$ alloys - an inference
confirmed by further explicit resistivity calculations. Both of these alloy
systems belong to the same branch of the famous Slater-Pauling plot and we
postulate that other alloy systems from this branch should show a similar
behavior. Our calculations show that the appearance of the plateau in the
resistivity can be attributed to the dominant contribution of minority spin
electrons to the conductivity which is nearly unaffected by increase in Cr/V
concentration $x$ and we remark that this minority spin electron feature is
also responsible for the simple linear variation of the average moment in the
Slater-Pauling plot for these materials.
