Cayley graph

 ( Group Theory ) 

In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group,

is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs make them particularly good candidates for constructing expander graphs.

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Definition Let $G$ be a group and $S$ be a generating set of $G$. The Cayley graph $\backslash Gamma\; =\; \backslash Gamma(G,S)$ is an Edge coloring directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405.

• Each element $g$ of $G$ is assigned a vertex: the vertex set of $\backslash Gamma$ is identified with $G.$
• Each element $s$ of $S$ is assigned a color $c\_s$.
• For every $g\; \backslash in\; G$ and $s\; \backslash in\; S$, there is a directed edge of color $c\_s$ from the vertex corresponding to $g$ to the one corresponding to $gs$.

Not every convention requires that $S$ generate the group. If $S$ is not a generating set for $G$, then $\backslash Gamma$ is disconnected and each connected component represents a coset of the subgroup generated by $S$.

If an element $s$ of $S$ is its own inverse, $s\; =\; s^\{-1\},$ then it is typically represented by an undirected edge.

The set $S$ is often assumed to be finite, especially in geometric group theory, which corresponds to $\backslash Gamma$ being locally finite and $G$ being finitely generated.

The set $S$ is sometimes assumed to be Symmetric set ($S\; =\; S^\{-1\}$) and not containing the group identity element. In this case, the uncolored Cayley graph can be represented as a simple undirected graph.

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Examples

• Suppose that $G=\backslash Z$ is the infinite cyclic group and the set $S$ consists of the standard generator 1 and its inverse (−1 in the additive notation); then the Cayley graph is an infinite path.
• Similarly, if $G=\backslash Z\_n$ is the finite cyclic group of order $n$ and the set $S$ consists of two elements, the standard generator of $G$ and its inverse, then the Cayley graph is the cycle graph $C\_n$. More generally, the Cayley graphs of finite cyclic groups are exactly the .
• The Cayley graph of the direct product of groups (with the cartesian product of generating sets as a generating set) is the cartesian product of the corresponding Cayley graphs.. Thus the Cayley graph of the abelian group $\backslash Z^2$ with the set of generators consisting of four elements $(\backslash pm\; 1,0),(0,\backslash pm\; 1)$ is the infinite grid graph on the plane $\backslash R^2$, while for the direct product $\backslash Z\_n\; \backslash times\; \backslash Z\_m$ with similar generators the Cayley graph is the $n\; \backslash times\; m$ finite grid on a torus.

• A Cayley graph of the dihedral group $D\_4$ on two generators $a$ and $b$ is depicted to the left. Red arrows represent composition with $a$. Since $b$ is Cayley table, the blue lines, which represent composition with $b$, are undirected. Therefore the graph is mixed: it has eight vertices, eight arrows, and four edges. The Cayley table of the group $D\_4$ can be derived from the group presentation $$\backslash langle\; a,\; b\; \backslash mid\; a^4\; =\; b^2\; =\; e,\; a\; b\; =\; b\; a^3\; \backslash rangle.$$ A different Cayley graph of $D\_4$ is shown on the right. $b$ is still the horizontal reflection and is represented by blue lines, and $c$ is a diagonal reflection and is represented by pink lines. As both reflections are self-inverse the Cayley graph on the right is completely undirected. This graph corresponds to the presentation $$\backslash langle\; b,\; c\; \backslash mid\; b^2\; =\; c^2\; =\; e,\; bcbc\; =\; cbcb\; \backslash rangle.$$
• The Cayley graph of the free group on two generators $a$ and $b$ corresponding to the set $S\; =\; \backslash \{a,\; b,\; a^\{-1\},\; b^\{-1\}\backslash \}$ is depicted at the top of the article, with $e$ being the identity. Travelling along an edge to the right represents right multiplication by $a,$ while travelling along an edge upward corresponds to the multiplication by $b.$ Since the free group has no relations, the Cayley graph has no cycles: it is the 4-regular graph infinite tree. It is a key ingredient in the proof of the Banach–Tarski paradox.
• More generally, the Bethe lattice or Cayley tree is the Cayley graph of the free group on $n$ generators. A presentation of a group $G$ by $n$ generators corresponds to a surjective homomorphism from the free group on $n$ generators to the group $G,$ defining a map from the Cayley tree to the Cayley graph of $G$. Interpreting graphs Topology as one-dimensional simplicial complexes, the simply connected infinite tree is the universal cover of the Cayley graph; and the kernel of the mapping is the fundamental group of the Cayley graph.
• A Cayley graph of the discrete Heisenberg group $$\backslash left\backslash \{\; \backslash begin\{pmatrix\}$$

1 & x & z\\
0 & 1 & y\\
0 & 0 & 1\\
     

\end{pmatrix},\ x,y,z \in \Z\right\} is depicted to the right. The generators used in the picture are the three matrices $X,\; Y,\; Z$ given by the three permutations of 1, 0, 0 for the entries $x,\; y,\; z$. They satisfy the relations $Z\; =\; XYX^\{-1\}Y^\{-1\},\; XZ\; =\; ZX,\; YZ\; =\; ZY$, which can also be understood from the picture. This is a nonabelian group infinite group, and despite being embedded in a three-dimensional space, the Cayley graph has four-dimensional volume growth.

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Characterization The group $G$ acts on itself by left multiplication (see Cayley's theorem). This may be viewed as the action of $G$ on its Cayley graph. Explicitly, an element $h\backslash in\; G$ maps a vertex $g\backslash in\; V(\backslash Gamma)$ to the vertex $hg\backslash in\; V(\backslash Gamma).$ The set of edges of the Cayley graph and their color is preserved by this action: the edge $(g,gs)$ is mapped to the edge $(hg,hgs)$, both having color $c\_s$. In fact, all automorphisms of the colored directed graph $\backslash Gamma$ are of this form, so that $G$ is isomorphic to the symmetry group of $\backslash Gamma$.

The left multiplication action of a group on itself is simply transitive, in particular, Cayley graphs are vertex-transitive. The following is a kind of converse to this:

To recover the group $G$ and the generating set $S$ from the unlabeled directed graph $\backslash Gamma$, select a vertex $v\_1\backslash in\; V(\backslash Gamma)$ and label it by the identity element of the group. Then label each vertex $v$ of $\backslash Gamma$ by the unique element of $G$ that maps $v\_1$ to $v.$ The set $S$ of generators of $G$ that yields $\backslash Gamma$ as the Cayley graph $\backslash Gamma(G,S)$ is the set of labels of out-neighbors of $v\_1$. Since $\backslash Gamma$ is uncolored, it might have more directed graph automorphisms than the left multiplication maps, for example group automorphisms of $G$ which permute $S$.

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Elementary properties

• The Cayley graph $\backslash Gamma(G,S)$ depends in an essential way on the choice of the set $S$ of generators. For example, if the generating set $S$ has $k$ elements then each vertex of the Cayley graph has $k$ incoming and $k$ outgoing directed edges. In the case of a symmetric generating set $S$ with $r$ elements, the Cayley graph is a regular graph of degree $r.$
• Cycles (or closed walks) in the Cayley graph indicate relations among the elements of $S.$ In the more elaborate construction of the Cayley complex of a group, closed paths corresponding to relations are "filled in" by . This means that the problem of constructing the Cayley graph of a given presentation $\backslash mathcal\{P\}$ is equivalent to solving the Word Problem for $\backslash mathcal\{P\}$.
• If $f:\; G\text{'}\backslash to\; G$ is a surjective group homomorphism and the images of the elements of the generating set $S\text{'}$ for $G\text{'}$ are distinct, then it induces a covering of graphs $$\backslash bar\{f\}:\; \backslash Gamma(G\text{'},S\text{'})\backslash to\; \backslash Gamma(G,S),$$ where $S\; =\; f(S\text{'}).$ In particular, if a group $G$ has $k$ generators, all of order different from 2, and the set $S$ consists of these generators together with their inverses, then the Cayley graph $\backslash Gamma(G,S)$ is covered by the infinite regular tree of degree $2k$ corresponding to the free group on the same set of generators.
• For any finite Cayley graph, considered as undirected, the vertex connectivity is at least equal to 2/3 of the degree of the graph. If the generating set is minimal (removal of any element and, if present, its inverse from the generating set leaves a set which is not generating), the vertex connectivity is equal to the degree. The edge connectivity is in all cases equal to the degree.See Theorem 3.7 of
  László Babai (1995). 9780444823465, Elsevier. . ISBN 9780444823465
• If $\backslash rho\_\{\backslash text\{reg\}\}(g)(x)\; =\; gx$ is the left-regular representation with $|G|\backslash times\; |G|$ matrix form denoted $\backslash rho\_\{\backslash text\{reg\}\}(g)$, the adjacency matrix of $\backslash Gamma(G,S)$ is $A\; =\; \backslash sum\_\{s\backslash in\; S\}\; \backslash rho\_\{\backslash text\{reg\}\}(s)$.
• Every group character $\backslash chi$ of the group $G$ induces an eigenvector of the adjacency matrix of $\backslash Gamma(G,S)$. The associated eigenvalue is $$\backslash lambda\_\backslash chi=\backslash sum\_\{s\backslash in\; S\}\backslash chi(s),$$ which, when $G$ is Abelian, takes the form $$\backslash sum\_\{s\backslash in\; S\}\; e^\{2\backslash pi\; ijs/|G|\}$$ for integers $j\; =\; 0,1,\backslash dots,|G|-1.$ In particular, the associated eigenvalue of the trivial character (the one sending every element to 1) is the degree of $\backslash Gamma(G,S)$, that is, the order of $S$. If $G$ is an Abelian group, there are exactly $|G|$ characters, determining all eigenvalues. The corresponding orthonormal basis of eigenvectors is given by $v\_j\; =\; \backslash tfrac\{1\}\{\backslash sqrt$}\begin{pmatrix} 1 & e^{2\pi ij/|G|} & e^{2\cdot 2\pi ij/|G|} & e^{3\cdot 2\pi ij/|G|} & \cdots & e^{(|G|-1)2\pi ij/|G|}\end{pmatrix}. It is interesting to note that this eigenbasis is independent of the generating set $S$. More generally for symmetric generating sets, take $\backslash rho\_1,\backslash dots,\backslash rho\_k$ a complete set of irreducible representations of $G,$ and let $\backslash rho\_i(S)\; =\; \backslash sum\_\{s\backslash in\; S\}\; \backslash rho\_i(s)$ with eigenvalue set $\backslash Lambda\_i(S)$. Then the set of eigenvalues of $\backslash Gamma(G,S)$ is exactly $\backslash bigcup\_i\; \backslash Lambda\_i(S),$ where eigenvalue $\backslash lambda$ appears with multiplicity $\backslash dim(\backslash rho\_i)$ for each occurrence of $\backslash lambda$ as an eigenvalue of $\backslash rho\_i(S).$

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Schreier coset graph If one instead takes the vertices to be right cosets of a fixed subgroup $H,$ one obtains a related construction, the Schreier coset graph, which is at the basis of coset enumeration or the Todd–Coxeter process.

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Connection to group theory Knowledge about the structure of the group can be obtained by studying the adjacency matrix of the graph and in particular applying the theorems of spectral graph theory. Conversely, for symmetric generating sets, the spectral and representation theory of $\backslash Gamma(G,S)$ are directly tied together: take $\backslash rho\_1,\backslash dots,\backslash rho\_k$ a complete set of irreducible representations of $G,$ and let $\backslash rho\_i(S)\; =\; \backslash sum\_\{s\; \backslash in\; S\}\; \backslash rho\_i(s)$ with eigenvalues $\backslash Lambda\_i(S)$. Then the set of eigenvalues of $\backslash Gamma(G,S)$ is exactly $\backslash bigcup\_i\; \backslash Lambda\_i(S),$ where eigenvalue $\backslash lambda$ appears with multiplicity $\backslash dim(\backslash rho\_i)$ for each occurrence of $\backslash lambda$ as an eigenvalue of $\backslash rho\_i(S).$

The genus of a group is the minimum genus for any Cayley graph of that group.

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Geometric group theory For infinite groups, the Coarse structure of the Cayley graph is fundamental to geometric group theory. For a finitely generated group, this is independent of choice of finite set of generators, hence an intrinsic property of the group. This is only interesting for infinite groups: every finite group is coarsely equivalent to a point (or the trivial group), since one can choose as finite set of generators the entire group.

Formally, for a given choice of generators, one has the word metric (the natural distance on the Cayley graph), which determines a metric space. The coarse equivalence class of this space is an invariant of the group.

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Expansion properties When $S\; =\; S^\{-1\}$, the Cayley graph $\backslash Gamma(G,S)$ is $|S|$-regular, so spectral techniques may be used to analyze the expander graph of the graph. In particular for abelian groups, the eigenvalues of the Cayley graph are more easily computable and given by $\backslash lambda\_\backslash chi\; =\; \backslash sum\_\{s\backslash in\; S\}\; \backslash chi(s)$ with top eigenvalue equal to $|S|$, so we may use Cheeger's inequality to bound the edge expansion ratio using the spectral gap.

Representation theory can be used to construct such expanding Cayley graphs, in the form of Kazhdan property (T). The following statement holds:Proposition 1.12 in

For example the group $G\; =\; \backslash mathrm\{SL\}\_3(\backslash Z)$ has property (T) and is generated by elementary matrices and this gives relatively explicit examples of expander graphs.

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Integral classification An integral graph is one whose eigenvalues are all integers. While the complete classification of integral graphs remains an open problem, the Cayley graphs of certain groups are always integral. Using previous characterizations of the spectrum of Cayley graphs, note that $\backslash Gamma(G,S)$ is integral iff the eigenvalues of $\backslash rho(S)$ are integral for every representation $\backslash rho$ of $G$.

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Cayley integral simple group A group $G$ is Cayley integral simple (CIS) if the connected Cayley graph $\backslash Gamma(G,S)$ is integral exactly when the symmetric generating set $S$ is the complement of a subgroup of $G$. A result of Ahmady, Bell, and Mohar shows that all CIS groups are isomorphic to $\backslash mathbb\{Z\}/p\backslash mathbb\{Z\},\; \backslash mathbb\{Z\}/p^2\backslash mathbb\{Z\}$, or $\backslash mathbb\{Z\}\_2\; \backslash times\; \backslash mathbb\{Z\}\_2$ for primes $p$. It is important that $S$ actually generates the entire group $G$ in order for the Cayley graph to be connected. (If $S$ does not generate $G$, the Cayley graph may still be integral, but the complement of $S$ is not necessarily a subgroup.)

In the example of $G=\backslash mathbb\{Z\}/5\backslash mathbb\{Z\}$, the symmetric generating sets (up to graph isomorphism) are

• $S\; =\; \backslash \{1,4\backslash \}$: $\backslash Gamma(G,S)$ is a $5$-cycle with eigenvalues $2,\; \backslash tfrac\{\backslash sqrt\{5\}-1\}\{2\},\backslash tfrac\{\backslash sqrt\{5\}-1\}\{2\},\backslash tfrac\{-\backslash sqrt\{5\}-1\}\{2\},\backslash tfrac\{-\backslash sqrt\{5\}-1\}\{2\}$
• $S\; =\; \backslash \{1,2,3,4\backslash \}$: $\backslash Gamma(G,S)$ is $K\_5$ with eigenvalues $4,\; -1,-1,-1,-1$

The only subgroups of $\backslash mathbb\{Z\}/5\backslash mathbb\{Z\}$ are the whole group and the trivial group, and the only symmetric generating set $S$ that produces an integral graph is the complement of the trivial group. Therefore $\backslash mathbb\{Z\}/5\backslash mathbb\{Z\}$ must be a CIS group.

The proof of the complete CIS classification uses the fact that every subgroup and homomorphic image of a CIS group is also a CIS group.

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Cayley integral group A slightly different notion is that of a Cayley integral group $G$, in which every symmetric subset $S$ produces an integral graph $\backslash Gamma(G,S)$. Note that $S$ no longer has to generate the entire group.

The complete list of Cayley integral groups is given by $\backslash mathbb\{Z\}\_2^n\backslash times\; \backslash mathbb\{Z\}\_3^m,\backslash mathbb\{Z\}\_2^n\backslash times\; \backslash mathbb\{Z\}\_4^n,\; Q\_8\backslash times\; \backslash mathbb\{Z\}\_2^n,S\_3$, and the dicyclic group of order $12$, where $m,n\backslash in\; \backslash mathbb\{Z\}\_\{\backslash ge\; 0\}$ and $Q\_8$ is the quaternion group. The proof relies on two important properties of Cayley integral groups:

• Subgroups and homomorphic images of Cayley integral groups are also Cayley integral groups.
• A group is Cayley integral iff every connected Cayley graph of the group is also integral.

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Normal and Eulerian generating sets Given a general group $G$, a subset $S\; \backslash subseteq\; G$ is normal if $S$ is closed under conjugation by elements of $G$ (generalizing the notion of a normal subgroup), and $S$ is Eulerian if for every $s\; \backslash in\; S$, the set of elements generating the cyclic group $\backslash langle\; s\; \backslash rangle$ is also contained in $S$. A 2019 result by Guo, Lytkina, Mazurov, and Revin proves that the Cayley graph $\backslash Gamma(G,S)$ is integral for any Eulerian normal subset $S\; \backslash subseteq\; G$, using purely representation theoretic techniques.

The proof of this result is relatively short: given $S$ an Eulerian normal subset, select $x\_1,\backslash dots,\; x\_t\backslash in\; G$ pairwise nonconjugate so that $S$ is the union of the conjugacy classes $\backslash operatorname\{Cl\}(x\_i)$. Then using the characterization of the spectrum of a Cayley graph, one can show the eigenvalues of $\backslash Gamma(G,S)$ are given by $\backslash left\backslash \{\backslash lambda\_\backslash chi\; =\; \backslash sum\_\{i=1\}^t\; \backslash frac\{\backslash chi(x\_i)\; \backslash left|\backslash operatorname\{Cl\}(x\_i)\backslash right|\}\{\backslash chi(1)\}\backslash right\backslash \}$ taken over irreducible characters $\backslash chi$ of $G$. Each eigenvalue $\backslash lambda\_\backslash chi$ in this set must be an element of $\backslash mathbb\{Q\}(\backslash zeta)$ for $\backslash zeta$ a primitive $m^\{th\}$ root of unity (where $m$ must be divisible by the orders of each $x\_i$). Because the eigenvalues are algebraic integers, to show they are integral it suffices to show that they are rational, and it suffices to show $\backslash lambda\_\backslash chi$ is fixed under any automorphism $\backslash sigma$ of $\backslash mathbb\{Q\}(\backslash zeta)$. There must be some $k$ relatively prime to $m$ such that $\backslash sigma(\backslash chi(x\_i))\; =\; \backslash chi(x\_i^k)$ for all $i$, and because $S$ is both Eulerian and normal, $\backslash sigma(\backslash chi(x\_i))\; =\; \backslash chi(x\_j)$ for some $j$. Sending $x\backslash mapsto\; x^k$ bijects conjugacy classes, so $\backslash operatorname\{Cl\}(x\_i)$ and $\backslash operatorname\{Cl\}(x\_j)$ have the same size and $\backslash sigma$ merely permutes terms in the sum for $\backslash lambda\_\backslash chi$. Therefore $\backslash lambda\_\backslash chi$ is fixed for all automorphisms of $\backslash mathbb\{Q\}(\backslash zeta)$, so $\backslash lambda\_\backslash chi$ is rational and thus integral.

Consequently, if $G=A\_n$ is the alternating group and $S$ is a set of permutations given by $\backslash \{\; (12i)^\{\backslash pm\; 1\}\; \backslash \}$, then the Cayley graph $\backslash Gamma(A\_n,S)$ is integral. (This solved a previously open problem from the Kourovka Notebook.) In addition when $G\; =\; S\_n$ is the symmetric group and $S$ is either the set of all transpositions or the set of transpositions involving a particular element, the Cayley graph $\backslash Gamma(G,S)$ is also integral.

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History Cayley graphs were first considered for finite groups by Arthur Cayley in 1878. Max Dehn in his unpublished lectures on group theory from 1909–10 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of the word problem for the fundamental group of surfaces with genus ≥ 2, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point.

Max Dehn (2025). 9781461291077, Springer-Verlag. ISBN 9781461291077

Translated from the German and with introductions and an appendix by John Stillwell, and with an appendix by Otto Schreier.

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See also

• Vertex-transitive graph
• Generating set of a group
• Lovász conjecture
• Cube-connected cycles
• Algebraic graph theory
• Cycle graph (algebra)

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Notes

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External links

• Cayley diagrams

Categories: Group Theory, Permutation Groups, Graph Families, Application-specific Graphs, Geometric Group Theory

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