Abstract
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 pairof parameters k ≥ 2 and g ≥ 3, improve the lower bounds on the order of the smallest ( k , g ) vertex-transitive graphs forcertain 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.
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