One of the most well known applications of representation theory in combinatorics is the explicit construction of Expander graphs by Margulis (using Kazhdan's property (T)). See any book about expanders (such as Lubotzky's) for example.The combinatorial statement in this case can be thought as having bounded Cheeger constant, or equivalently bounding the Rayleigh quotient.
EDIT - in order to address the last sentence, a non-constructive proof (by the probabilistic method) was known before Margulis' construction (probably due to Kolmogorov, but usually formally attributed to Pinsker). Since Margulis, a whole industry of explicit constructions has been developed, most notably with applications of arithmetic combinatorics in the last few years (what's known as the Bourgain-Gamburd method), although one might argue whether arithmetic combinatorics is part of combinatorics or harmonic analysis.