Asymptotic aspects of Cayley graphs

Details Asymptotic aspects of Cayley graphs Asymptotic aspects of Cayley graphs

2011-08-07

arxiv.org/abs/1108.1571v1

Mathematics

Combinatorics

16 pages, 6 figures

Scientific paper

Arising from complete Cayley graphs $\Gamma_n$ of odd cyclic groups $\Z_n$, an asymptotic approach is presented on connected labeled graphs whose vertices are labeled via equally-multicolored copies of $K_4$ in $\Gamma_n$ with adjacency of any two such vertices whenever they are represented by copies of $K_4$ in $\Gamma_n$ sharing two equally-multicolored triangles. In fact, these connected labeled graphs are shown to form a family of graphs of largest degree 6 and diameter asymptotically of order $|V|^{1/3}$, properties shared by the initial member of a collection of families of Cayley graphs of degree $2m\geq 6$ with diameter asymptotically of order $|V|^{1/m}$, where $3\leq m\in\Z$.

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