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13.
Small vertex-transitive graphs of given degree and girth
Jajcay, Robert ; Širan, Jozef
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 ▫$k \geq 2$▫ and ▫$g \geq 3$▫, 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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