We study when, and how compactly, a finite connected graph (G) embeds isometrically into a Cayley graph of a finite abelian group. The classical theory of partial cubes answers this for isometric subgraphs of hypercubes through the Djokovic-Winkler relation (\theta); we extend the question to the full family of abelian Cayley graphs, whose hosts may carry composite generators and cyclic factors of any order. We introduce an involutive edge relation (\varphi), defined by two simultaneous distance equalities, which coincides with (\theta) exactly on partial cubes and remains informative beyond them, together with an oriented relation (\Phi) for non-involutive hosts, where generator classes are constrained to be partial permutations rather than matchings.The central result is a quotient labeling theorem: for any partition of the edge set into candidate generator classes, the most generic consistent vertex labeling is the quotient of the free module on the classes by the lattice of signed cycle-class incidences, computed by the Smith normal form; the binary case is its reduction modulo two. We prove that the finest partition always yields an isometric labeling, that compactifying the resulting universal group is itself an instance of the same quotient construction, and that the whole construction is algorithmic and certifiable. Worked examples include the triangle, the Petersen graph (embedding into the Clebsch graph of order 16), the Pappus graph (a 1024-fold compaction), and the diamond (a non-diagonal fold). Sharp dimension bounds and an exhaustive census of small graphs are developed in a companion paper. 2020 MSC: 05C12, 05C25, 20K01, 05C50
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
We consider a number of examples of groups together with an infinite conjugation-invariant generating set, including the free group with the generating set of all separable elements, surface groups with the generating set of all non-filling curves, mapping class groups and outer automorphism groups of free groups with the generating sets of all reducible elements, and groups with suitable actions on Gromov hyperbolic spaces with a generating set of elliptic elements. Building on the work of Brandenbursky–Gal–Kędra–Marcinkowski, in these Cayley graphs, we show that there are quasi-isometrically embedded copies of
\mathbb{Z}^{m}
for all
m\geq1
. A corollary is that these Cayley graphs have infinite asymptotic dimension. By additionally building a new subsurface projection analogue for the free-splitting graph, which is valued in the above Cayley graph of the free group and may be of independent interest, we are able to recover Sabalka–Savchuk’s result that the edge-splitting graph of the free group has quasi-isometrically embedded copies of
{\mathbb{Z}}^{m}
for all
m\geq1
.
Sabine Chu, G. Domat, Christine Gao et al.· Groups, Geometry, and Dynami...· 0 citations
Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.
Vishal Gupta, A. Iosevich, J. Iosevich et al.· 0 citations
We introduce the $\textit{expanded core graph}$, which records minimal unbounded domains and their axial directions, for two classes of hierarchically hyperbolic group structures modeled on compact special groups and mapping class groups. Our main structural result shows that every embedding of a right-angled Artin group, after replacing its standard generators by positive powers, factors through an intermediate RAAG generated by suitably supported axial elements; for the class modeled on compact special groups, this intermediate RAAG is quasi-isometrically embedded. We show that its extension graph embeds into the expanded core graph. This yields a Kim--Koberda-type obstruction to RAAG embeddings and a complete embedding criterion when the rank is at most two. For the standard HHG structure on a mapping class group, the expanded core graph is the disjointness graph of essential curves, while for natural rich-family structures on a RAAG it recovers the extension graph. We also establish permanence results under finite direct products and the standard relatively hyperbolic construction.
The orientable genus polynomial of a graph counts its cellular embeddings by genus. For finite simple $2$-connected cubic graphs it is a cycle-matroid invariant: $M(G)\cong M(H)$ implies $\Gamma_G=\Gamma_H$. The adjacency spectrum and the genus polynomial are incomparable: neither determines the other. We exhibit connected cubic graphs on $16$ vertices sharing the adjacency spectrum, spanning-tree count, girth, diameter, vertex and edge connectivity, automorphism-group order, and cycle counts through length $10$, yet with pairwise distinct genus polynomials. Splitting the expected face count at twice the girth explains the difference: short faces are spectral, long faces are not. We construct an explicit infinite family of connected cospectral cubic pairs $(G_t,H_t)$ on $14+2t$ vertices whose minimum genera differ. We also compute the genus polynomials of all $7,875,918$ connected cubic graphs through $22$ vertices and derive from short-cycle counts a deterministic lower bound on the minimum genus.
A graph category is a category with a set of graphs or similar structures (such as, directed graphs, signed graphs, etc.) playing the role of objects, and an appropriate notion of homomorphism playing the role of morphisms. The characterization of multiplicative objects are important open problems in categories of undirected and directed graphs. While the recent disproving of the Hedetniemi's conjecture due to Shitov (Ann. Math. 2019), which claimed that all complete graphs are multiplicative, provided a breakthrough in the study of multiplicative undirected graphs, the characterization of multiplicative undirected graphs remains known only for cycles, circular cliques $K_{{n/k}}$ where ${n/k} \in (2,4]$, complete graphs, and graphs whose each edge is part of at most one $4$-cycle. Similarly, whether a given directed graph is multiplicative or not is known only for some oriented paths, oriented cycles, and transitive tournaments. We study multiplicative graphs in the category of directed graphs where pushable homomorphism plays the role of morphism. We provide full multiplicativity characterization for directed bipartite graphs, oriented cycles, and transitive tournaments. As a consequence we find new (infinite) classes of non-multiplicative directed graphs in the usual directed graphs category. We also resolve an open question posed by Das \textit{et al.} (CALDAM 2026) related to the existence of exponential directed graphs with respect to pushable homomorphisms, and use our solution as a tool for our proofs.
S. Das, Moritz Muhlenthaler, Sagnik Sen et al.· 0 citations
