--- title: 'Generalized Token Graphs: Theory & Extensions' url: https://www.emergentmind.com/topics/generalized-token-graph type: topic --- # Generalized Token Graphs: Theory & Extensions Generalized token graph denotes a family of constructions extending the classical \(k\)-token graph \(F_k(G)\), in which vertices represent token configurations on a host graph and adjacencies encode admissible token moves. In the classical setting, the vertices are the \(k\)-subsets of \(V(G)\), and two configurations are adjacent when one token moves along one edge to an unoccupied vertex. Recent work uses the term for several non-equivalent extensions: \(F_k^m(G)\), where \(m\) tokens move simultaneously; token digraphs \(F_k(D)\), where the host object is directed; supertoken graphs \(F_k^s(G)\) and \(\mathcal F_k(G)\), which allow repeated occupancy or distinguish token identities; and, in graph representation learning, a reversible graph-tokenization interface \(\Phi(G)=T(\sigma(G))\) that maps a labeled graph to a discrete token sequence without loss of information [2509.01773] [2410.20189] [2604.04845] [2604.06164] [2603.11099]. The phrase therefore refers less to a single canonical graph and more to a program of extending token-based representations of graph structure. ## 1. Classical foundation and taxonomy For a simple graph \(G=(V,E)\) on \(n\) vertices and an integer \(1\le k\le n\), the classical \(k\)-token graph \(F_k(G)\) has vertex set \[ V(F_k(G))=\{A\subseteq V(G): |A|=k\}, \] and two \(k\)-subsets are adjacent exactly when their symmetric difference is a pair of adjacent vertices of \(G\). Equivalently, a vertex of \(F_k(G)\) is a configuration of \(k\) indistinguishable tokens on distinct vertices of \(G\), and adjacency corresponds to moving exactly one token along one edge while the other \(k-1\) tokens remain fixed [2012.00808]. The same construction is also called the symmetric \(k\)-th power, and \(F_k(K_n)\) is the Johnson graph \(J(n,k)\) [2310.16929]. This classical model is the common base point for later generalizations. In the directed setting, when the host digraph \(D\) is replaced by an undirected graph \(G\), the \(k\)-token digraph construction collapses precisely to the undirected \(k\)-token graph \(F_k(G)\) [2410.20189]. In the multiplicity-based setting, the prohibition on repeated occupancy is relaxed. In the graph-learning setting, the “token” terminology is repurposed to mean discrete symbols or continuous embeddings derived from graph structure rather than vertices of a reconfiguration graph. | Construction | Distinguishing feature | Source | |---|---|---| | \(F_k(G)\) | one indistinguishable token moves to an unoccupied neighbor | [2012.00808] | | \(F_k^m(G)\) | \(m\) tokens move along \(m\) edges | [2509.01773] | | \(F_k(D)\) | host object is a digraph; adjacency is oriented | [2410.20189] | | \(F_k^s(G)\), \(F^s_{k\times1}(G)\), \(\mathcal F_k(G)\) | repeated occupancy allowed; colored and uncolored variants | [2604.04845] [2604.06164] | | \(\Phi(G)=T(\sigma(G))\) | graph serialized into a reversible token sequence | [2603.11099] | A common misconception is that “generalized token graph” names a unique object. The literature does not support that view. Instead, it comprises several extensions that preserve the token-configuration intuition while changing the admissible moves, occupancy constraints, host category, or even the meaning of “token.” ## 2. Simultaneous moves: the generalized token graph \(F_k^m(G)\) The paper “Generalized Token Graphs” defines \(F_k^m(G)\) for integers \(1\le m\le k\) as the graph whose vertices correspond to configurations of \(k\) indistinguishable tokens placed at distinct vertices of \(G\), where two configurations are adjacent whenever one configuration can be reached from the other by moving \(m\) tokens along \(m\) edges of \(G\). When \(m=1\), the usual token graph \(F_k(G)\) is recovered [2509.01773]. This extension changes the combinatorics of connectivity and parity in a way that differs sharply from the classical case. For \(F_2^2(G)\), if \(G\) is non-bipartite, then \(F_2^2(G)\) is connected if and only if \(G\) does not contain two or more leaves all sharing the same neighbor. Moreover, if \(G\) has exactly \(k\ge2\) leaves attached to a single vertex \(v\), then \(F_2^2(G)\) has one nontrivial connected component plus \(\binom{k}{2}\) isolated vertices. Likewise, \(F_2^2(G)\) is bipartite if and only if \(G\) is a disjoint union of paths; equivalently, \(G\) contains no cycle and no vertex of degree at least \(3\). These criteria show that classical implications do not transfer unchanged once simultaneous moves are permitted. The same work analyzes several graph parameters. For complete bipartite hosts, \(F_2^2(K_{m,n})\) satisfies \(\omega(F_2^2(K_{m,n}))=mn\) and \(\chi(F_2^2(K_{m,n}))=mn\), while \[ \alpha(F_2^2(K_{m,n}))=\max\left\{\binom{n}{2},\binom{m}{2}\right\}+1 \] and \(\gamma(F_2^2(K_{m,n}))=3\). For cycles, \[ \omega(F_2^2(C_n))= \begin{cases} 4,& n=4,\\ 3,& n\in\{3,6\},\\ 2,& \text{otherwise,} \end{cases} \qquad \chi(F_2^2(C_n))= \begin{cases} 4,& n=4,\\ 3,& n\neq 4. \end{cases} \] The paper also gives \(\alpha(F_2^2(P_n))=\lfloor n/2\rfloor\lceil n/2\rceil\), identifies \(F_2^2(C_{2k+1})\cong C_{2k+1}\square P_k\), and shows that for odd cycles \[ \operatorname{Aut}(F_2^2(C_n))\cong D_n\times S_2. \] Automorphism behavior illustrates another nontrivial departure from the base graph. Every automorphism of \(G\) induces one of \(F_k^m(G)\), so \(\operatorname{Aut}(G)\hookrightarrow \operatorname{Aut}(F_k^m(G))\); however, equality need not hold. In the diamond example \(D=K_4-e\), \(\operatorname{Aut}(D)\cong S_2\times S_2\), whereas \(\operatorname{Aut}(F_2^2(D))\cong S_4\). This shows that generalized token graphs can amplify symmetry rather than merely inherit it. ## 3. Repeated occupancy and colored tokens: supertoken frameworks A second major line of generalization allows more than one token at a vertex. In “On Generalized Token Graphs,” the indistinguishable-token supertoken graph \(F_k^s(G)\) is defined for a simple graph \(G\), a token count \(k\), and a per-vertex capacity \(1\le s\le k\). A vertex of \(F_k^s(G)\) is an unordered multiset \(A=(\alpha_1,\dots,\alpha_k)\) in which no element of \(V\) appears more than \(s\) times, equivalently an integer vector \((x_1,\dots,x_n)\) with \(0\le x_i\le s\) and \(x_1+\cdots+x_n=k\). Two vertices are adjacent if and only if their multisets differ by moving a single token along one edge of \(G\). The same paper defines the colored supertoken graph \(F^s_{k\times1}(G)\), where tokens are distinguishable and vertices are ordered \(k\)-tuples subject to the same occupancy constraint [2604.04845]. This framework unifies several classical constructions. The usual token graph is recovered as \(F_k(G)=F_k^1(G)\). When \(s=k\) and tokens are indistinguishable, one gets the reduced power \(F_k^k(G)\). When \(s=k\) and tokens are distinguishable, one gets the Cartesian product \(G\Box G\Box\cdots\Box G\) with \(k\) factors. The order formulas are given explicitly: for indistinguishable tokens, \[ |V(F_k^s(G))| = \sum_{i=0}^{\lfloor k/(s+1)\rfloor} (-1)^i\binom{n}{i}\binom{n+k-1-i(s+1)}{n-1}, \] and for distinguishable tokens the paper provides a refined inclusion–exclusion expression \(h(n,k,s)\). It also gives corresponding edge-count formulas. For connectivity, if \(G\) is connected and \(k\le n-1\), all nontrivial supertoken graphs \(F_k^s(G)\) and \(F^s_{k\times1}(G)\) are connected, except \(F_n^1(G)\). A related unrestricted-multiplicity construction appears in “On supertoken graphs,” which defines the \(k\)-supertoken graph \(\mathcal F_k(G)\) as the graph whose vertices are all multisets of size \(k\) drawn from \(V(G)\), represented as nonnegative integer vectors \(x=(x_1,\dots,x_n)\) with \(\sum_i x_i=k\). Two vertices \(x,y\) are adjacent exactly when \(y=x-e_i+e_j\) for some edge \(\{i,j\}\in E(G)\) [2604.06164]. Its order and size are \[ |V(\mathcal F_k(G))|=\binom{n+k-1}{k}, \qquad |E(\mathcal F_k(G))|=m\binom{n+k-2}{k-1}. \] The same paper proves \[ G\cong \mathcal F_1(G)\subseteq \mathcal F_k(G)\subseteq \mathcal F_{k+1}(G), \] \[ \operatorname{diam}(\mathcal F_k(G))=k\,\operatorname{diam}(G), \qquad \operatorname{rad}(\mathcal F_k(G))\le k\,\operatorname{rad}(G), \] and shows that the eigenvalues of \(\mathcal F_{k-1}(G)\) interlace those of \(\mathcal F_k(G)\). For graph invariants, \(\mathcal F_k(G)\) satisfies \[ \omega(\mathcal F_k(G))=\omega(G), \qquad \chi(\mathcal F_k(G))=\chi(G), \] and the paper develops lower bounds and exact values for \(\alpha(\mathcal F_k(G))\), including explicit formulas for \(\mathcal F_2(C_n)\). It also constructs the \(p\)-augmented \(2\)-token graphs of cycles \(F_2^p(C_n)\), with \[ |V(F_2^p(C_n))|=\binom{n}{2}+pn, \qquad |E(F_2^p(C_n))|=n^2+2pn, \] and, for odd \(n\), \[ \rho(F_2^p(C_n))=4\cos\!\left(\frac{\pi}{n+2p}\right). \] The supertoken literature makes clear that “generalization” can refer not to more complicated move rules, but to relaxing the exclusion constraint itself. This suggests that occupancy constraints, token identity, and move locality are largely orthogonal design choices. ## 4. Directed token graphs and related complexity questions For a digraph \(D\) of order \(n\) and \(1\le k\le n-1\), the \(k\)-token digraph \(F_k(D)\) has vertex set \[ V(F_k(D))=\{A\subseteq V(D): |A|=k\}, \] and arc set consisting of ordered pairs \((A,B)\) such that \(A\setminus B=\{a\}\), \(B\setminus A=\{b\}\), and \((a,b)\) is an arc of \(D\). Equivalently, a single token moves along one arc of \(D\) at each step, keeping the other \(k-1\) tokens fixed [2410.20189]. This directed version refines several undirected correspondences. If \(D\) is strongly connected then so is \(F_k(D)\), and in fact \(F_k(D)\) is strongly connected if and only if \(D\) is. More generally, if \(C_1,\dots,C_t\) are the strongly connected components of \(D\) in topological order and a configuration \(A\) places \(k_j\) tokens in \(C_j\), then the strongly connected component of \(F_k(D)\) containing \(A\) is isomorphic to the Cartesian product \(\prod_{j:k_j>0}F_{k_j}(C_j)\). The paper also proves that \(F_k(D)\) is acyclic if and only if \(D\) is acyclic. Kernel theory behaves more subtly. If \(D\) has no oriented odd cycle then neither does \(F_k(D)\); hence, by the theorem of von Neumann–Morgenstern, every such \(F_k(D)\) has a unique kernel. But the presence of odd cycles breaks any simple lifting principle: the paper gives small examples of \(D\) with a kernel but \(F_2(D)\) having none, and vice versa. It further shows that deciding whether \(F_2(D)\) admits a kernel is NP-complete, via an adaptation of Chvátal’s reduction from NAE-3SAT. Several directed invariants are preserved or controlled sharply. The oriented girth and circumference satisfy \[ g(F_k(D))=g(D), \qquad c(F_k(D))\ge c(D), \] with stronger lower bounds on \(c(F_k(D))\) when \(c(D)\ge5\) and \(2\le k\le n-3\). Eulerianity is preserved exactly: \[ D \text{ is Eulerian } \Longleftrightarrow F_k(D) \text{ is Eulerian.} \] For bidirected clique number, \[ \omega^+(F_k(D))=\min\{\omega^+(D),\max\{n-k+1,k+1\}\}, \] and for dichromatic number, \[ \vec\chi(F_k(D))\le \vec\chi(D). \] Combined with the bound of Cordero-Michel and Galeana-Sánchez, \[ \vec\chi(D)\le \left\lceil\frac{c(D)-1}{g(D)-1}\right\rceil+1, \] the same uniform bound follows for \(F_k(D)\). A related but distinct reconfiguration literature generalizes token jumping rather than token graphs. Under \((k,d)\)-Token-Jumping, one may move several tokens at once provided each travels distance at most \(d\). For connected graphs, the minimal distances guaranteeing reconfigurability of every pair of solutions are \(d_{\min}=1\) for Vertex Cover, \(d_{\min}=2\) for Dominating Set, and \(d_{\min}=3\) for Independent Set [2411.12582]. This is not a token-graph construction in the strict sense, but it belongs to the same reconfiguration paradigm. ## 5. Spectral and algebraic structure inherited from classical token graphs Much of the generalized-token-graph literature is organized around invariants first understood for the classical \(F_k(G)\). For Laplacian spectra, if \(1\le h\le k\le n/2\), then \[ \operatorname{Spec} L(F_h(G))\subseteq \operatorname{Spec} L(F_k(G)), \] and the inclusion can be proved by lifting eigenvectors through the inclusion matrices \(W_{h,k}\) or via the matrix identity \[ L(F_h(G))=W_{h,k}^{\top}L(F_k(G))W_{h,k} \] in the formulation of the paper [2012.00808]. The same work proves that \(F_k(\overline G)\) and \(F_k(G)\) are coupled through the Johnson graph: \[ L_k+\overline L_k=L(J(n,k)), \] and formulates the conjecture that for every graph \(G\) and every \(k=1,\dots,|V(G)|-1\), \[ \alpha(F_k(G))=\alpha(G). \] The algebraic framework behind this complement relation is developed further in “On two algebras of token graphs.” There, the corrected commutativity statement \[ L_k\overline L_k=\overline L_kL_k \] is used to define the local algebra \[ \mathcal L(G)=\langle L_k,\overline L_k\rangle, \] a unital commutative matrix algebra containing the Bose–Mesner algebra of the Johnson graph \(J(n,k)\). The same paper then defines a global algebra \(\mathcal A(n,k)\), spanned by the adjacency matrices \(A_e\) attached to single edges \(e\subseteq K_n\), with \[ \dim \mathcal A(n,k)=\binom{n}{2}, \] and shows that \(\mathcal A(n,k)\) contains the adjacency and Laplacian matrices of the \(k\)-token graph of any graph \(G\) on \(n\) vertices [2403.18800]. Adjacency spectra exhibit similarly rigid patterns. For walk-regular \(G\), “On the spectra and spectral radii of token graphs” proves the exact formula \[ \rho(F_k(G))=k\,\rho^{\,k-1}(G), \] where \(\rho^{\,k-1}(G)\) is the common spectral radius of all \((k-1)\)-vertex-deleted subgraphs. For \(k=2\), this becomes \(\rho(F_2(G))=2\,\rho(G\setminus u)\) for any vertex \(u\). When \(G\) is distance-regular, the partition of \(F_2(G)\) by distances between the two occupied vertices is equitable, and the resulting quotient matrix yields explicit eigenvalues of \(F_2(G)\). The paper also proposes generalized Aldous-type monotonicity conjectures for the “new” eigenvalues \(\alpha_k(G)\) and \(\rho_k(G)\) that appear when passing from \(F_{k-1}(G)\) to \(F_k(G)\) [2310.16929]. These results concern the classical exclusion model rather than every generalized variant. Still, they provide the main template for later work on generalized constructions: commutativity, interlacing, quotient matrices, and preserved connectivity parameters are the dominant analytic tools. ## 6. Terminological extension in graph representation learning In graph machine learning, “Generalized Token Graph” has acquired a separate meaning. “Graph Tokenization for Bridging Graphs and Transformers” defines a framework in which a labeled graph \(G=(V,E)\) is serialized by a reversible, deterministic map \[ \sigma:G\to S,\qquad S=(s_0,e_1,s_1,e_2,\dots,e_K,s_K), \] guided by global node–edge–node \(3\)-gram frequencies \(F(p)\), and then compressed by Byte Pair Encoding to obtain \[ \Phi(G)=T(\sigma(G)). \] The paper proves injectivity up to isomorphism, states that the final vocabulary size is \(|V_T|=|\Sigma|+K\), reports that \(K\approx 2\,000\) typically, and gives a token composition of approximately \(7\%\) atomic tokens, \(28\%\) small substructures, \(42\%\) medium substructures, and \(23\%\) larger motifs [2603.11099]. It further reports state-of-the-art results on 14 benchmark datasets, “10× shorter sequences after BPE,” and “\(\sim2\)–\(3\times\) faster Transformer training,” while requiring “No change to standard Transformer or BERT.” A second line keeps the token notion continuous rather than discrete. TEA-GLM treats graph token embeddings as a small set of continuous vectors aligned with the token embeddings of an LLM. Its GNN is pretrained with an instance-wise contrastive loss \(L_{\mathrm{ins}}\) and a feature-wise loss \(L_{\mathrm{fea}}\), combined as \[ L_{\rm pretrain}=\frac12(L_{\mathrm{ins}}+L_{\mathrm{fea}}), \] after projection into the principal-component subspace of the LLM token embedding matrix. A frozen GNN representation \(u\in\mathbb R^{F_L}\) is then mapped by a single affine layer to \[ H_{\mathrm{token}}\in\mathbb R^{K\times F_L}, \] and these instance-specific soft tokens are inserted into a unified instruction prompt for zero-shot node classification and link prediction [2408.14512]. The paper reports zero-shot node-classification results of PubMed \(84.8\%\pm1.0\) and Cora \(20.2\%\pm1.4\) in the citation-domain transfer setting, best AUCs on several link-prediction benchmarks, and ablations in which removing \(L_{\mathrm{fea}}\) degrades transfer while removing graph token embeddings collapses zero-shot performance nearly to random. TokenGT pushes the terminology in yet another direction by treating all nodes and edges as independent tokens and feeding them, with orthonormal node identifiers and type identifiers, into a vanilla Transformer. With sufficiently expressive token embeddings, a single Transformer layer with \(15\) heads can approximate any equivariant linear map \(L_{2\to2}\), and the resulting architecture is at least as expressive as a \(2\)-IGN, hence strictly more powerful than standard message-passing GNNs [2207.02505]. On PCQM4Mv2, the paper reports validation MAE \(0.0910\) for TokenGT with Laplacian identifiers, compared with \(0.1083\) for GIN-VN and \(0.0864\) for Graphormer. The alignment problem created by graph tokenization is analyzed explicitly in RGLM. That work models Graph-Tokenizing LLMs as LLMs consuming a fixed-length graph-token sequence \(v=(P_\phi\circ F)(\mathcal G)\), argues that text-only instruction tuning yields only implicit graph–text alignment, and proves \[ \mathcal I(x_t;s^G\mid s^T_{