  In this work we are concerned with the problem of achieving max-min fairness
in Gaussian parallel channels with respect to a general performance function,
including channel capacity or decoding reliability as special cases. As our
central results, we characterize the laws which determine the value of the
achievable max-min fair performance as a function of channel sharing policy and
power allocation (to channels and users). In particular, we show that the
max-min fair performance behaves as a specialized version of the Lovasz
function, or Delsarte bound, of a certain graph induced by channel sharing
combinatorics. We also prove that, in addition to such graph, merely a certain
2-norm distance dependent on the allowable power allocations and used
performance functions, is sufficient for the characterization of max-min fair
performance up to some candidate interval. Our results show also a specific
role played by odd cycles in the graph induced by the channel sharing policy
and we present an interesting relation between max-min fairness in parallel
channels and optimal throughput in an associated interference channel.
