  In this paper, we have considered the optimization of the $M$-ary quadrature
amplitude modulation (MQAM) constellation size to minimize the bit energy
consumption under average bit error rate (BER) constraints. In the computation
of the energy expenditure, the circuit, transmission, and retransmission
energies are taken into account. A combined log-normal shadowing and Rayleigh
fading model is employed to model the wireless channel. The link reliabilities
and retransmission probabilities are determined through the outage
probabilities under log-normal shadowing effects. Both single-hop and multi-hop
transmissions are considered. Through numerical results, the optimal
constellation sizes are identified. Several interesting observations with
practical implications are made. It is seen that while large constellations are
preferred at small transmission distances, constellation size should be
decreased as the distance increases. Similar trends are observed in both fixed
and variable transmit power scenarios. We have noted that variable power
schemes can attain higher energy-efficiencies. The analysis of energy-efficient
modulation design is also conducted in multi-hop linear networks. In this case,
the modulation size and routing paths are jointly optimized, and the analysis
of both the bit energy and delay experienced in the linear network is provided.
