Dimension and Order Bounds for Isometric Embeddings of Graphs into Abelian Cayley Graphs, and the Abelian Dividend
We investigate the minimum size of finite abelian Cayley graphs that admit an isometric embedding of a finite connected graph. While every connected graph on n vertices embeds isometrically into a binary Cayley graph of dimension at most n-1, the smallest possible abelian host has remained largely unexplored. We establish fundamental lower bounds showing that every binary host has dimension at least max(diam(G), floor(log2 n)), whereas every finite abelian host has order at least max(n, 2^diam(G)). Moreover, we prove that the minimum host order equals n if and only if G is itself an abelian Cayley graph. Exact binary dimensions are obtained for several important graph families. Hypercubes, complete graphs of order 2^k, and even cycles attain the lower bound. For stars we prove k_min(K1,q)=floor(log2 q)+1 using maximum sum-free sets, yielding an exponential improvement over the naive and isometric dimensions. For odd cycles we prove k_min(Cm)=m-1 for all m<17 and reduce the general case to a cyclic-interval lemma, showing that the universal upper bound is tight. Our computational contribution is a certified exhaustive census of all 995 connected graphs with 2<=n<=7 vertices under general abelian compactifications. The data reveal an"abelian dividend": 569 graphs (57 percent) admit a strictly smaller abelian host than the best binary host, 707 (71 percent) admit an optimal host containing a cyclic factor Zm with m>2, and only 17 graphs attain the theoretical order floor max(n,2^diam(G)). These results demonstrate that compact non-binary abelian hosts are typical rather than exceptional, while binary hosts remain the universal worst-case construction. 2020 MSC:05C12, 05C25, 05C30, 11B75, 20K01