Quite a bit of work over many decades has gone into exploring respiratory rhythm-generation mechanisms. These studies have established an important role for the pre-Botzinger (pBC) complex in the mammalian brainstem and has investigated properties of single pBC neurons and their synaptic interactions. I will present work arising from efforts to understand how synchronous bursts of activity emerge across the network of pBC respiratory neurons. This work includes some computational approaches to systematically study how burst synchrony depends on network properties, some analytical approaches to estimate impacts of the prevalence of architectural motifs on the spread of activity in a network, and some rigorous analysis of graphicality and graph enumeration that are relevant to testing the motif-based ideas computationally.