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Small vertex-transitive graphs of given degree and girth

We investigate the basic interplay between the small k-valent vertex-transitive graphs of girth g and the (k;g)-cages, the smallest k-valent graphs of girth g. We prove the existence of k-valent Cayley graphs of girth g for every pair of parameters k 2 and g 3, improve the lower bounds on the order of the smallest (k;g) vertex-transitive graphs for certain families with prime power girth, and generalize the construction of Bray, Parker and Rowley that has yielded several of the smallest known (k;g)-graphs.