Solution group representations as quantum symmetries of graphs

In 2019, Aterias et al. constructed pairs of quantum isomorphic, non-isomorphic graphs from linear constraint systems. This article deals with quantum automorphisms and quantum isomorphisms of colored versions of those graphs. We show that the quantum automorphism group of such a colored graph is the dual of the homogeneous solution group of the underlying linear constraint system. Given a vertex-and edge-colored graph with certain properties, we construct an uncol-ored graph that has the same quantum automorphism group as the colored graph we started with. Using those results, we obtain the first-known example of a graph that has quantum symmetry and finite quantum auto-morphism group. Furthermore, we construct a pair of quantum isomorphic, non-isomorphic graphs that both have no quantum symmetry.


Introduction
Generalizing Mermin's magic square [12], linear system games were introduced by Cleve and Mittal in [7].It was then shown by Cleve, Liu and Slofstra [6], that perfect quantum strategies for those games are related to the representations of the associated solution group; a finitely presented group with generators and relations reflected by the underlying linear system.
Together with his collaborators, the first author [1] constructed quantum isomorphic, but non-isomorphic graphs from linear system games having perfect quantum strategies but no such classical strategies.In this case, a quantum isomorphism corresponds to a perfect quantum strategy of the isomorphism game which was also introduced in [1].
Quantum automorphism groups of graphs were first introduced by Banica [2] and Bichon [5] to obtain examples of quantum permutation groups.They are generalizations of the automorphism group of a graph in the framework of Woronowicz' compact matrix quantum groups.An interesting question to ask is which quantum permutation groups can be realized as the quantum automorphism group of some graph.This question has for example been considered in [4].
In this work, we give a correspondence between representations of solution groups and quantum isomorphisms/quantum automorphisms of colored versions of the graphs appearing in [1].More specifically, we will see that the quantum automorphism groups of the colored versions of those graphs are given by the dual of the homogeneous solution group of the underlying linear constraint system.Furthermore, representations of the non-homogeneous solution group provide quantum isomorphisms between the colored versions of the graphs.
Additionally, we will discuss a decoloring procedure for vertex-and edge-colored graphs which does not change the quantum automorphism group for certain graphs.This procedure decolors the vertices by adding paths of different lengths to vertices of different colors.In a second step, we get rid of the edge-colors by subdividing the colored edges and then adding paths of different lengths to the newly added vertices.
The previous results allow us to construct two explicit examples of graphs that were not known before: First, we construct a graph that has quantum symmetry and finite quantum automorphism group.Second, we obtain a pair of graphs that are quantum isomorphic and non-isomorphic, where both graphs additionally do not have quantum symmetry.
The article is structured as follows.In Section 2, we briefly discuss compact quantum groups and give the definition of the quantum automorphism group of a vertex-and edgecolored graph.In Section 3, we define colored versions of the graphs appearing in [1].Here we prove that the dual of the solution group is the quantum automorphism group of such a graph.Section 4 deals with decoloring the graphs from the previous section without changing the quantum automorphism group.In Section 5, we provide (uncolored) graphs whose quantum automorphism group is the dual of a solution group.We in particular obtain a graph with quantum symmetry and finite quantum automorphism group, see Corollary 5.6.Finally, we discuss quantum isomorphisms of the colored graphs in Section 6.Here, we present our example of a pair of graphs that are quantum isomorphic but non-isomorphic, where both graphs do not have quantum symmetry, see Corollary 6.10.

Preliminaries
We start with the definition of a compact quantum group, see [20].Throughout this article, we write A ⊗ B for the minimal tensor product of the C * -algebras A and B.
We then write G ∼ = H.Definition 2.3.Let G = (C(G), ∆ G ) be a compact quantum group.We say that G is finite if the C * -algebra C(G) is finite-dimensional.
The following example can for example be found in [11,Example 4.5].
We will now define compact matrix quantum groups, a subclass of compact quantum groups.
Definition 2.5 ([19]).A compact matrix quantum group G is a pair (C(G), u), where C(G) is a unital C * -algebra and u = (u ij ) ∈ M n (C(G)) is a matrix such that • the matrix u and its transpose u T are invertible.
The matrix u is usually called fundamental representation of G.
A very important example is the quantum symmetric group, the quantum analogue of the symmetric group.It was defined by Wang in [17].
Definition 2.6.The quantum symmetric group S + n = (C(S + n ), u) is the compact matrix quantum group, where Note that a matrix u = (u ij ) i,j∈ [n] with entries from a nontrivial unital C * -algebra satisfying , as in the definition above, is known as a magic unitary.
Quantum automorphism groups of finite graphs were defined in [2], [5].We give a more general definition in the following, for vertex-and edge-colored graphs.Note that edgecolorings are similar to distances in finite quantum metric spaces, as considered in [3].For us, graphs are undirected and do not have loops nor multiple edges.A colored graph is a graph G with vertex set V and edge set E along with a coloring function c : V ∪ E → S for some set S. We refer to c(x) as the color of the vertex/edge x.To be specific, we will sometimes refer to colored graphs as vertex-and edge-colored graphs.Furthermore, we will also consider edge-colored graphs where the coloring function is defined only on the edge set E, i.e., edges but not vertices are colored.In either case we will use E c to refer to the set of edges of color c.Also, A Gc will denote the adjacency matrix of the edge color c, i.e., (A Gc ) ij = 1 if (i, j) ∈ E c and (A Gc ) ij = 0 otherwise.Definition 2.7.Let G be a vertex-and edge-colored graph.The quantum automorphism group Qut(G) is the compact matrix quantum group (C(Qut(G)), u), where C(Qut(G)) is the universal C * -algebra with generators u ij , i, j ∈ V (G) and relations where ( 4) is nothing but k u ik (A Gc ) kj = k (A Gc ) ik u kj for all i, j ∈ V (G) and all edge-colors c.
It is not immediately obvious that this defines a compact matrix quantum group.By Definition 2.5 we have to show that u and u T are invertible as well as that the comultiplication n is a compact matrix quantum group.The first equation follows for example from [15,Lemma 2.1.2].For the second equation, we take vertices i, j with c(i) = c(j).We have ∆(u ij ) = k u ik ⊗ u kj .Note that there is no k ∈ V (G) with c(k) = c(i) and c(k) = c(j), since we assumed c(i) = c(j).We deduce u ik ⊗ u kj = 0 for all k and thus ∆(u ij ) = 0. Summarizing, we see that Qut(G) is indeed a compact matrix quantum group.
The following lemma gives relations that are equivalent to Relation (4), see [15, Proposition 2.1.3].Lemma 2.8.Let u ij , 1 ≤ i, j ≤ n, be the generators of C(S + n ).Then Relation (4) is equivalent to the relations for some edge-color c.

Colored graphs whose quantum automorphism group is the dual of a solution group
In the following definition we use 1 to denote the identity element of a group.1.
If b = 0, then we refer to Γ(M, b) as the homogeneous solution group of the system M x = 0, and define this the same as above except that we add the relation γ = 1.This is equivalent to removing γ from the list of generators, changing the righthand side of the equation in (3) to 1, and removing items (4) and (5).We will sometimes also use Γ 0 (M ) to denote this group.
We will denote by S k (M ) the set {i ∈ [n] : M ki = 1}, often writing simply S k when M is clear from context.We use ±1 S to denote the set of functions α : S → {1, −1}, and will typically write α i instead of α(i).We will also use ±1 S 0 to denote the subset of such functions satisfying i∈S α i = 1, and similarly use ±1 S 1 for the set of such functions satisfying i∈S α i = −1.
, and these partition the vertex set.For any l, k ∈ [m] such that S l ∩ S k = ∅, the graph G contains all (non-loop) edges between V l and V k (thus each V k induces a complete subgraph).Given an edge e between adjacent vertices (l, α) and (k, β), the color of e, denoted c(e), is equal to the function α△β ∈ ±1 S l ∩S k defined as (α△β) i = α i β i .
Note that α△β = β△α and so the edge colors defined above really are edge colors and not arc (directed edge) colors.Remark 3.3.According to the above definition, it is possible for two edges e and e ′ between pairs of vertices (l, α), (k, β) and (l ′ , α ′ ), (k ′ , β ′ ) to be colored the same color even if l is not equal to either l ′ nor k ′ , i.e., the edges are between different pairs of subsets V l , V k and V l ′ , V k ′ .This can happen since it is possible that S l ∩ S k = S l ′ ∩ S k ′ and α△β = α ′ △β ′ .However, we do wish such edges to be distinguished by the (quantum) automorphism group of G(M, b), i.e., u (l,α),(l ′ α ′ ) u (k,β),(k ′ β ′ ) = 0 where u is the fundamental representation of Qut(G(M, b)) 1 .This could be done explicitly by defining the color of the edge e (for instance) to be c(e) = ({l, k}, α△β).However, this is redundant for our purposes since such edges e and e ′ as described above are already distinguished by the (quantum) automorphism group due to the vertex colors of the endpoints of the edges.In other words, the edges between V l and V k can be thought of as being implicitly colored distinctly from those between V l ′ and V k ′ whenever {l, k} = {l ′ , k ′ }.Remark 3.4.We also note that in order to reduce the total number of colors used, for each pair l, k ∈ [m], we can choose one color α ∈ ±1 S l ∩S k and replace these edges with non-edges.Moreover, instead of coloring the edges between V l and V k with functions from ±1 S l ∩S k , we can simply use {1, . . ., 2 |S l ∩S k | − 1}, and this will not change Qut(G) as explained in the previous remark.In fact we will need to do this for some of our results later on.
Example 3.5.Consider the linear system M x = b, where Then the graph G(M, b) is the one given in Figure 1. ( (1, −1, −1) (1, 1, −1) We used Remarks 3.3 and 3.4 to reduce the number of colors needed in the graph.
where each u (k) is a magic unitary.Furthermore, u Proof.If l = k, then c((l, α)) = l = k = c((k, β)) and thus u (l,α),(k,β) = 0.This shows that u has the block form given in the lemma statement.Moreover, each diagonal block must be a magic unitary since u is a magic unitary.Now fix k ∈ [m].Pick α, β ∈ ±1 S k b k arbitrarily.Suppose that α, β ∈ ±1 S k b k are such that α△β = α△ β.Note that since all these functions are elements of ±1 S k b k , the operation △ is simply the pointwise product, and therefore we have that α△α = β△β.In particular, this implies that the color of the edge between (k, α) and (k, α) is equal to the color of the edge between (k, β) and (k, β ′ ) if and only if β ′ = β.Therefore, u Thus the operator u Thus the set of operators appearing in any row (and similarly any column) of Since the entries of any given row of a magic unitary commute, and every row of u (k) contains the same set of entries, we have that all entries of u (k) commute.Remark 3.7.Let Γ = Γ(M, 0) be the homogeneous solution group of the system M x = 0.By Example 2.4, the group C * -algebra C * (Γ) can be written as Note that we slightly abuse the notation by writing x i instead of u x i .
Proof.The proof is structured as follows: we use the universal properties of C * (Γ) and C(Qut(G)) respectively to first obtain a * -homomorphism ϕ 1 : C * (Γ) → C(Qut(G)), and then obtain a *homomorphism ϕ 2 : C(Qut(G)) → C * (Γ).We will then show that ϕ 1 and ϕ 2 are inverses of each other thus proving that they are in fact isomorphisms.Lastly, we will prove that ϕ 1 intertwines the coproducts as described in the theorem statement.
Step 1: Construction of a * -homomorphism For this step, we will construct elements y i of C(Qut(G)) that satisfy the relations of the generators x i of C * (Γ) given in Remark 3.7.This proves that there is a * -homomorphism ϕ 1 from C * (Γ) to C(Qut(G)) such that ϕ 1 (x i ) = y i .Later we will see that ϕ 1 is in fact an isomorphism.
Let u be the fundamental representation of Qut(G).By Lemma 3.6, u = k∈[m] u (k) where each u (k) is a magic unitary indexed by α,β := u (k,α),(k,β) depends only on k and the value of α△β ∈ ±1 S k 0 .Thus, as in the proof of Lemma 3.6, for each δ ∈ ±1 S k 0 we let u α,β such that α△β = δ.Note that for any 0 and thus every row/column of u (k) contains the same set of operators and Now we define y α for all k ∈ [m] and i ∈ S k .We first aim to show that Consider the edges between the subsets V l and V k .For each δ ∈ ±1 S l ∩S k , let A δ be the adjacency matrix of the graph consisting of the edges of G colored δ.Further, let B δ be the submatrix of A δ consisting of the rows indexed by V l and columns indexed by V k .In other words, , and similarly define B + and B − .Then Since u must commute with each A δ by definition of Qut(G), it must also commute with both A + and A − .This implies that u (l) . Considering the (l, α), (k, β) entry of both sides of the former equation, we see that Note that for every term in the above sums, we have that α where α = α△α ′ , and similarly for u Doing the same for u (l) B − = B − u (k) yields the same equation but with α i β i replaced with −α i β i and combining these proves that i .

So we have shown that the value of y (k)
i does not depend on k, and thus we will simply denote this operator by y i .Now note that since y i is a linear combination (with real coefficients) of the operators u (k) α for k ∈ [m] such that i ∈ S k , and these operators are entries of the magic unitary u, we have that y * i = y i .Also, by Equation ( 6) we have that u β = 0 for α = β and thus Thus the y i satisfy relation (1) from Definition 3.1.Next we will show that relation (2) of Definition 3.1 holds, i.e., that y i y j = y j y i if there exists k ∈ [m] such that i, j ∈ S k .Suppose that i, j, k are as described.Then y i = y (k) i and y j = y (k) j are both linear combinations of the entries of u (k) which pairwise commute by Lemma 3.6.Therefore y i and y j commute as desired.
Lastly, we must show that relation (3) of Definition 3.1 holds.Recall that we are trying to show that C * (Γ 0 (M )) ∼ = C(Qut(G)), i.e., we are in the homogeneous solution group case.Thus we must show that i∈S k y i = 1 for all k ∈ [m].We have that α are pairwise orthogonal, when we expand the above product all cross terms disappear and we obtain as desired.Therefore the elements y i ∈ C(Qut(G)) for i ∈ [n] satisfy the relations of the generators of C * (Γ) and thus there exists a * -homomorphism Step 2: Construction of a * -homomorphism For this step, we will construct elements v (l,α),(k,β) of C * (Γ) that satisfy all the relations of the generators u (l,α),(k,β) of C(Qut(G)).This proves that there is a * -homomorphism . Later we will see that ϕ 2 is an isomorphism.
, and that Note that the latter implies that p + i and p − i are orthogonal, i.e., p + i p − i = 0. We will abuse notation somewhat and write p α i i to denote p + i whenever α i = +1 and similarly for p − i when α are indeed well defined since the commutativity of the elements p ± i for i ∈ S k follows from the commutativity of the x i for i ∈ S k (i.e., relation (2) from Definition 3.1).Since it is the product of pairwise commuting projections, we have that v This also implies that for a fixed k ∈ [m] the projections v α are pairwise orthogonal.Now suppose that α ∈ ±1 S k 1 , i.e., that i∈S k α i = −1.Then, using the easily checked fact that x i p α i i = α i p α i i , we have that Thus v Combining this with Equation ( 7), we have that Next, for all k ∈ [m] and α, β δ where δ = α△β.Lastly, define v to the the matrix indexed by V (G) such that We aim to show that v satisfies all of the relations satisfied by the fundamental representation u of Qut(G).First, v is a magic unitary since all of its entries are projections and the sum of its  This means there exists j ∈ S l ∩ S k such that α j α ′ j = β j β ′ j .Since all these terms are in {+1, −1}, we have that δ j = α j β j = α ′ j β ′ j = δ ′ j .This implies that p So we have shown that v is a magic unitary satisfying all of the relations satisfied by u and thus by the universal property of C(Qut(G)), there exists a * -homomorphism Step 3: Showing that ϕ 1 and ϕ 2 are inverses of each other.
We first show that ϕ 2 • ϕ 1 : C * (Γ) → C * (Γ) is the identity.Since both ϕ 1 and ϕ 2 are *homomorphisms, it suffices to show that ϕ 2 • ϕ 1 acts as the identity on the generators To see that the final expression above is equal to x i , we will show that multiplying it by x i yields 1.First, note that since x i p α i i = α i p α i i , we have that Thus, making use of Equation ( 9) in the last equality, we have that ) is the inverse of x i , which is its own inverse by definition.Therefore ϕ 2 • ϕ 1 (x i ) = x i and thus ϕ 2 • ϕ 1 is the identity map on C * (Γ).Now we will show that is the identity map.As above, it suffices to show that ϕ 1 •ϕ 2 acts as the identity on all entries of the fundamental representation u.Other than 0, the entries of u are precisely the elements u where we have used the fact that α to denote the final expression of Equation (10).We wish to show that Since the y i satisfy all of the relations that the x i satisfy, the same argument as for the v Suppose that α = β.Then there exists j ∈ S k such that α j = β j and therefore where the penultimate equality comes from the fact that α j = β j implies that α j β j = −1.Using the above we have that This completes the proof that α and thus that ϕ 1 • ϕ 2 is the identity on C(Qut(G)).Combining with the above, this proves that ϕ 1 : C * (Γ) → C(Qut(G)) and ϕ 2 : C(Qut(G)) → C * (Γ) are isomorphisms of these C * -algebras which are inverse to each other. Step Since ϕ 1 and the coproducts are * -homomorphisms, it suffices to prove the identity on the generators, i.e., that First, let us determine ∆ G (u α,δ△α and thus Since we have already shown that (ϕ 1 ⊗ ϕ 1 ) • ∆ Γ (x i ) = y i ⊗ y i , this completes the proof.

Constructing uncolored graphs that have the same quantum automorphism group as a given colored graph
In the following, we will describe a procedure that given certain vertex-and edge-colored graphs produces a decolored graph with isomorphic quantum automorphism group.This procedure is divided into two steps: We first decolor the vertices and then the edges.We start with a lemma known from [8, Lemma 3. The following lemma can be found in [14,Lemma 3.2].The distance d(v, w) between two vertices v, w ∈ V (G) is the length of a shortest path connecting v and w.Lemma 4.2.Let G be a finite graph and u vw , 1 ≤ v, w ≤ n be the generators of C(Qut(G)).If we have d(v, p) = d(w, q), then u vw u pq = 0.

Lemma 4.3. Let G be a finite graph and u the fundamental representation of
Proof.Let v, w ∈ V (G) and p ∈ V (G), d(v, p) = k as above.Using Relation (2) and Lemma 4.2, we get We conclude We will now define a edge-colored graph G ′ from the vertex-and edge-colored graph G. Definition 4.5.Let G be a vertex -and edge-colored graph.Depending on the color c, attach a path of length n c ∈ N 0 to every vertex colored c, where n c 1 = n c 2 for colors c 1 = c 2 and then decolor the vertices of the graph.We choose one of the edge-colors of G and let the edges in the paths all have this edge-color.We denote this new edge-colored (but not vertex-colored) graph by G ′ .
The next lemma gives a description of the fundamental representation of Qut(G ′ ).We will see in the following theorem that the quantum automorphism groups of G and G ′ are isomorphic.
Example for the construction of the graph G ′ from G. We chose the edge-colors between the added paths to be black.
where additionally u v i w i = u v 0 w 0 for all i and u v i w i = 0 for c(v) = c(w), c(.) being the vertex-colors in the original graph G. Proof.
Step 1: It holds u v i w 0 = u v 0 w i = 0 for i = 0. We know that it holds deg(v i ) ∈ {1, 2} for vertices v i with i > 0. Since we have deg(w 0 ) ≥ 3 by assumption, we get u v i w 0 = 0 by Lemma 4.1.We similarly obtain u v 0 w i = 0.
Step 2: We have u v i w j = 0 for i = j, i, j > 0. First assume i < j.Then v 0 is a vertex with d(v i , v 0 ) = i and deg(v 0 ) ≥ 3. Since i < j, we know that the vertices q with d(w j , q) = i are in the path added to w and thus deg(q) ∈ {1, 2}.We deduce u v i w j = 0 by Lemma 4.3.The case i > j follows similarly.
Step 3: It holds since u v 1 p 0 = 0 and u v 0 w 2 = 0 by Step 1. Furthermore, for i ≥ 1, we have since it holds that u v i+1 w i−1 = 0 and u v i w i+2 = 0 by Step 1 or Step 2. Note that if there is no w i+2 , we still get u v i w i = u v i+1 w i+1 by a similar calculation, since then w i is the only neighbor of w i+1 .
Step 4: It holds Proof.Let u be the fundamental representation of Qut(G ′ ).As in Lemma 4.6, we denote by v i , 0 ≤ i ≤ n c(v) the vertices in the added path with d(v, v i ) = i and let Let c e be the color of the edges in the attached paths in G ′ .For colors c = c e , we get that By Lemma 4.6, we directly see that uA (11)).The adjacency matrix where B i is the (V i−1 ×V i )-matrix with (B i ) a i−1 b i = δ ab .Also by Lemma 4.6, we see that uA G ′ ce = A G ′ ce u implies u 0 A Gc e = A Gc e u 0 .Furthermore, since (u i ) v i w i = (u 0 ) v 0 w 0 for all i, we see that C(Qut(G ′ )) is generated by the entries of u 0 .We conclude that C(Qut(G ′ )) is generated by a magic unitary u 0 that fulfills u 0 A Gc = A Gc u 0 for all edge-colors c and (u 0 ) v 0 w 0 = 0 for c(v) = c(w) (see Lemma 4.6).This yields a surjective * -homomorphism ϕ : C(Qut(G)) → C(Qut(G ′ )), w ab → (u 0 ) ab , where w is the fundamental representation of Qut(G).
For the other direction, take the fundamental representation w of Qut(G) and build the matrix w ′ as follows where we put (w i ) a i b i = w a 0 b 0 .Note that w ′ is a magic unitary if and only if the matrices w i are magic unitaries.We compute b;nc(b)≥i where we used w ab = 0 for c(a) = c(b) and the fact that w is a magic unitary.Similarly, we get a;nc(a)≥i (w i ) a i b i = 1 and thus w ′ is a magic unitary.It remains to show that w ′ commutes with A G ′ c for all edge colors c.We deduce from ( 12) and ( 14) that wA Gc = A Gc w implies 13) and ( 14), we see that Note that we have by definition of w i and B i .We similarly get that w i+1 B t i+1 = B t i+1 w i is automatically fulfilled by definition of w i and B i .Since we have wA Gc e = A Gc e w by assumption, we obtain We will now deal with the edge-colors of G ′ .First, we need the following easy lemma.
Lemma 4.8.Let w ij , 1 ≤ i, j ≤ n be elements in a C * -algebra such that the matrix w = (w ij ) 1≤i,j≤n is a magic unitary.Then w ij w kl + w il w kj is a projection if and only if w ij w kl = w kl w ij and w il w kj = w kj w il .
Proof.Let w ij w kl + w il w kj be a projection.Since it is self-adjoint, we have Multiplying by w ij and w il , respectively, yields w ij w kl = w ij w kl w ij and w il w kj = w il w kj w il .But, by taking adjoints, this implies w ij w kl = w kl w ij and w il w kj = w kj w il .The other direction is clear, since w ij w kl + w il w kj is the sum of two orthogonal projections if w ij w kl = w kl w ij and w il w kj = w kj w il .
We will define a (uncolored) graph G ′′ from the edge-colored graph G ′ .In the next theorem, we will then see that for certain graphs, the quantum automorphism groups of G ′ and G ′′ are isomorphic.Let G be a graph and e = (u, v) ∈ E(G).We say that we subdivide e if we delete the edge e = (u, v) from G and add a vertex w as well as edges (u, w) and (w, v) to the graph.Definition 4.9.Let G be a vertex -and edge-colored graph.First decolor the vertices by applying the construction as in Definition 4.5 to obtain the edge-colored graph G ′ .We denote the color of the newly added edges in the graph G ′ by c 0 .We subdivide each colored edge with c(e) = c 0 and add a path of length m c to the subdivision, where m c 1 = m c 2 for colors c 1 = c 2 and then decolor the edges in the graph G ′ .We call this graph G ′′ .
Figure 3: Example for the construction of the graph G ′′ from G ′ .We chose black to be the edge-color c 0 .Lemma 4.10.Let G be a vertex -and edge-colored graph such that deg(v) ≥ 3 for all v ∈ V (G) and let G ′′ be the graph as in Definition 4.9.We denote the vertex that subdivided the edge e in G by e 0 and the vertices in the added path with d(e 0 , e i ) = i by e i , 1 ≤ i ≤ m c(e) .Furthermore where u e i f i = u vx u wy + u vy u wx for c(e) = c(f ), e = (v, w), f = (x, y) and u e i f i = 0 for c(e) = c(f ).
and thus u e 0 v 1 = 0 by Lemma 4.1.If n c(v 0 ) ≥ 2, then v j has at least one neighbor of degree 1 or 2. Since e 0 only has neighbors of degree ≥ 3, we get u e 0 v j = 0 by Lemma 4.3.
Assume i = j and i, j = 0. Let furthermore i < j.Then e 0 is a vertex with d(e i , e 0 ) = i and deg(e 0 ) = 3.Since i < j, we know that the vertices q with d(v j , q) = i are in the path and thus deg(q) ∈ {1, 2}.We deduce u e i v j = 0 by Lemma 4.3.The case i > j follows similarly.
It remains to show We know deg(q) ≤ 3 for vertices q with d(q, e i ) = i, since either q = e 0 or q is a vertex in the added path.We deduce u e i v i = 0 by Lemma 4.3.Now assume n c(v 0 ) = 0.If m c(e 0 ) = 0, then we know deg(e 0 ) = 2 and thus u e 0 v 0 = 0 by Lemma 4.1.If m c(e 0 ) > 0, then e 0 has a neighbor of degree 1 or 2. If v 0 has no neighbor of this degree, then u e 0 v 0 = 0 by Lemma 4.3.The vertex v 0 only has a neighbor of degree 2 if there exists a subdivision f 0 of some edge f = (v 0 , w 0 ) with m c(f ) = 0.Then, it holds If m c(e 0 ) = 1, then deg(e 1 ) = 1 = 2 = deg(f 0 ) which yields u e 1 f 0 = 0 by Lemma 4.1.If m c(e 0 ) ≥ 2, then e 1 has a neighbor e 2 with deg(e 2 ) ∈ {1, 2}.Since the neighbors v 0 , w 0 of f 0 have degree ≥ 3, we get u e 1 f 0 = 0 by Lemma 4.3.We deduce u e 0 v 0 = 0 in all those cases.
Step 2: It holds u e i f 0 = u e 0 f i = 0 for i = 0. We know that it holds deg(e i ) ∈ {1, 2} for vertices e i with i > 0. If m c(f 0 ) > 0, then we have deg(f 0 ) = 3 and get u e i f 0 = 0 by Lemma 4.1.Let m c(f 0 ) = 0.If m c(e 0 ) = 1, then deg(e 1 ) = 1 and thus u e 1 f 0 = 0 by Lemma 4.1.If m c(e 0 ) ≥ 2, then e i has a neighbor with degree 1 or 2. Since the neighbors of f 0 have degree ≥ 3 by assumption, we get u e i f 0 = 0 by Lemma 4.3.We similarly obtain u e 0 f i = 0.
Step 3: It holds u e i f j = 0 for i = j, i, j > 0. First assume i < j.Then e 0 is a vertex with d(e i , e 0 ) = i and deg(e 0 ) = 3.Since i < j, we know that the vertices q with d(f j , q) = i are in the path and thus deg(q) ∈ {1, 2}.We deduce u e i f j = 0 by Lemma 4.3.The case i > j follows similarly.
Step 4: It holds u e i f i = u e 0 f 0 for all i.We first show u e 0 f 0 = u e 1 f 1 .It holds since u e 1 p 0 = 0 by Step 1 and u e 0 f 2 = 0 by Step 2. Furthermore, for i ≥ 1, we have = u e i+1 f i+1 since we have u e i+1 f i−1 = 0 and u e i f i+2 = 0 by Step 2 or Step 3. Note that if there is no f i+2 , we still get u e i f i = u e i+1 f i+1 by a similar calculation, since then f i is the only neighbor of f i+1 .Step 6: It holds u e i f i = u vx u wy + u vy u wx for e = (v, w), f = (x, y).We have where we used u vf 1 = 0, u wf 1 = 0 by Step 1.Note that e 0 is the only vertex in E ′ 0 that is a common neighbor of v and w.Thus, we have (u vx + u vy )u kf 0 (u wx + u wy ) = 0 for all k = e 0 by Lemma 4.

By
Step 4, we obtain u e i f i = u vx u wy + u vy u wx .Remark 4.11.Note that the operator u (v,w)(x,y) = u vx u wy + u vy u wx does not depend on the order of (v, w) or (x, y), since u (v,w)(x,y) = u vx u wy + u vy u wx = u vy u wx + u vx u wy = u (v,w),(y,x) , u (v,w)(y,x) = u vy u wx + u vx u wy = u wx u vy + u wy u vx = u (w,v),(x,y) .
Before stating the theorem, we first need to define a quantum subgroup of the quantum automorphism group of the graph G ′ .It is straightforward to check that the comultiplication is a *-homomorphism.Definition 4.12.Let G be a vertex -and edge-colored graph and G ′ as in Definiton 4.5.We define Qut * c 0 (G ′ ) to be the compact matrix quantum group whose corresponding C * -algebra is generated by a magic unitary x with xA Gc = A Gc x for every edge color c and x ik x jl = x jl x ik for c((i, j)) = c((k, l)), c = c 0 , where c 0 is the edge-color we choose for the newly added edges in G ′ .Theorem 4.13.Let G be a vertex -and edge-colored graph such that deg(v) ≥ 3 for all v ∈ V (G).Let G ′′ be the graph as in Definition 4.9 and let Qut * c 0 (G ′ ) be the compact matrix quantum group as in Definition 4.12.Then there exists a * -isomorphism ϕ : Proof.Let u be the fundamental representation of Qut(G ′′ ).As in Lemma 4.10, we denote the vertex that subdivided the edge e in G by e 0 and the vertices in the added path with d(e 0 , e i ) = i by e i , 1 ≤ i ≤ m c(e) .Moreover, we denote by v i , 0 ≤ i ≤ n c(v) the vertices in the added path with d(v, v i ) = i.The adjacency matrix of G ′′ is of the form where A G 0 is the adjacency matrix of the edge-color that is not subdivided together with the paths from the construction of G ′ , B G ′′ is the matrix with (B G ′′ ) ve = 1 for v incident to e, 0 otherwise, , where w and u 0 are blocks in u (see (15)).Furthermore, since (u i ) e i f i = w ak w bl + w al w bk for e = (a, b), f = (k, l), we also have that C(Qut(G ′′ )) is generated by the entries of the matrix w.We will now show that w fulfills the relations of the generators of C(Qut * c 0 (G ′ )).We already know wA G ′ 0 = A G ′ 0 w.We will now show wA Gc = A Gc w for every edge color c = c 0 .By Lemma 2.8, this is equivalent to w ik w jl = 0 for c((i, j)) = c((k, l)) if (i, j) ∈ E ′ 0 or (k, l) ∈ E ′ 0 .We will show that the elements of w fulfill those relations.Let v ∈ V (G ′ ), (a, b) = e ∈ E ′ 0 .It holds Multiplying w va from left and right yields We deduce s;(v,s) / ∈E ′ 0 w va w sb w va = 0 and since all elements in the sum are positive, we get w va w sb w va = 0 for all v, s ∈ V (G ′ ) with (v, s) / ∈ E ′ 0 .We deduce w va w sb = 0 for all v, s ∈ V with (v, s) / ∈ E ′ 0 .Similarly, by using u 0 B t G ′′ = B t G ′′ w, we get w av w bs = 0 for all v, s ∈ V (G ′ ) with (v, s) / ∈ E ′ 0 .It remains to show w ik w jl = 0 for c(e) = c(f ), e = (i, j) ∈ E ′ 0 , f = (k, l) ∈ E ′ 0 .We know (u 0 ) ef = 0 for c(e) = c(f ) from Lemma 4.10.Then w ik w jl + w il w jk = (u 0 ) ef = 0 and by multiplying w ik and w il from the left, respectively, we get w ik w jl = 0 and w il w jk = 0. Thus, we have shown w ik w jl = 0 for c((i, j)) = c((k, l)) if (i, j) ∈ E ′ 0 or (k, l) ∈ E ′ 0 which is equivalent to wA Gc = A Gc w for every edge color c = c 0 .
Summarizing, we get wA Gc = A Gc w for every edge color c.By Lemma 4.10, we furthermore know that (u 0 ) ab = w ik w jl + w il w jk for a = (i, j), b = (k, l) and c(a) = c(b), c(a) = c 0 and thus w ik w jl + w il w jk is a projection.Then Lemma 4.8 yields w ij w kl = w kl w ij and w il w kj = w kj w il .Therefore, we get a surjective * -homomorphism ϕ : , where w ′ is the fundamental representation of Qut * c 0 (G ′ ).For the other direction, take the fundamental representation w ′ of Qut * c 0 (G ′ ) and build the matrix w ′′ as follows where we used w ′ ak w ′ bl = 0 for c((a, b)) = c((k, l)) and the fact that w ′ is a magic unitary.We get f ;nc(f )≥i (u ′ i ) f i e i = 1 similarly and therefore w ′′ is a magic unitary.It remains to show that w ′′ commutes with A G ′′ .Similar to equation (17), we see that and therefore 16) and the form of w ′′ , we see that which is true because of the following: We get the following corollary, which we will use in the next section to construct a graph with quantum symmetry and finite quantum automorphism group.

(Uncolored) graphs whose quantum automorphism group is the dual of a solution group
In this section, we will look at certain graphs from Definition 3.2, where we replace one of the edge colors by non-edges.Furthermore, we restrict to graphs coming from linear constraint systems as in the following definition.By using Theorem 3.8 and the decoloring procedure in Section 4, we will obtain (decolored) graphs whose quantum automorphism group is the dual of the corresponding solution group.
By the isomorphism above, we get that C(Qut(G ′′ * (M K 3,4 , 0))) is finite-dimensional and non-commutative which yields the assertion.

Quantum Isomorphisms
In this section we consider quantum isomorphisms between the colored graphs G(M, b) and G(M, b ′ ) for b = b ′ .In particular we give analogs of Lemma 3.6 and Theorem 3.8 for this case.From this we are able to obtain non-isomorphic colored graphs G(M, b) and G(M, b ′ ) that are quantum isomorphic but neither G(M, b) nor G(M, b ′ ) has quantum symmetry.Further, applying decoloring techniques as in Section 4 allows us to produce non-isomorphic uncolored graphs G and G ′ without quantum symmetry that are nevertheless quantum isomorphic.This appears to be the first known such example.
To define quantum isomorphism of colored graphs, we must provide a suitable generalization of the isomorphism game.The way to do this is quite natural.Definition 6.1 (Isomorphism game for colored graphs).Given colored graphs G and G ′ , with respective color functions c and c ′ , the (G, G ′ )-isomorphism game has as inputs and outputs for both Alice and Bob the set V (G) ∪ V (G ′ )2 .To win, upon receiving x ∈ V (G) Alice (respectively Bob) must respond with y ∈ V (G ′ ) and vice versa.If this condition is met, then there is a vertex g a ∈ V (G) that is either Alice's input or output, and there is similarly g b ∈ V (G) and g ′ a , g ′ b ∈ V (G ′ ).Alice and Bob then win if the following conditions are met: 3. (g a , g b ) is an edge of color c if and only if (g ′ a , g ′ b ) is an edge of color c.
We then say that two colored graphs G and G ′ are quantum isomorphic if there is a quantum strategy3 that wins the (G, G ′ )-isomorphism game with probability 1.In [10] it was shown that (uncolored) graphs G and G ′ are quantum isomorphic if and only if there exists a magic unitary u such that A G u = uA G ′ .Precisely the same proof applied to colored graphs gives the following: Proposition 6.2.Colored graphs G and G ′ with coloring functions c and c ′ are quantum isomorphic if and only if there exists a V (G) × V (G ′ ) magic unitary u such that c(g) = c ′ (g ′ ) implies u gg ′ = 0 for g ∈ V (G), g ′ ∈ V (G ′ ), and A Gc u = uA G ′ c for all edge colors c. i .
So we have shown that the value of y (k) i does not depend on k, and thus we will simply denote this operator by y i .Now note that since y i is a linear combination (with real coefficients) of the operators u Thus the y i satisfy relation (1) from Definition 6.5.Next we will show that relation (2) of Definition 6.5 holds, i.e., that y i y j = y j y i if there exists k ∈ [m] such that i, j ∈ S k .Suppose that i, j, k are as described.Then y i = y (k) i and y j = y (k) j are both linear combinations of the entries of u (k) which pairwise commute by Lemma 6.6.Therefore y i and y j commute as desired.
Lastly, we must show that relation α are pairwise orthogonal, when we expand the above product all cross terms disappear and we obtain by definition.Therefore the elements y i ∈ Iso(G, G ′ ) for i ∈ [n] satisfy the relations of the generators of A and thus there exists a *homomorphism ϕ 1 from A to Iso(G, G ′ ) such that ϕ 1 (x i ) = y i for all i ∈ [n].
Step 2: Construction of a * -homomorphism ϕ 2 : Iso(G, G ′ ) → A. This step is almost identical to Step 2 of the proof of Theorem 3.8, and so we omit it.We only remark that the biggest change is that when showing that v  8) gets an additional factor of (−1) b k +b ′ k in the first and last expressions.
Step 3: Showing that ϕ 1 and ϕ 2 are inverses of each other.This step is nearly identical to Step 3 of the proof of Theorem 3.8 and so we omit it.

Definition 3 . 1 .
Let M ∈ F m×n 2 and b ∈ F m 2 with b = 0.The solution group Γ(M, b) of the linear system M x = b is the group generated by elements x i for i ∈ [n] and an element γ satisfying the following relations:

Figure 1 :
Figure 1: The graph G(M, b) for M and b as above.The vertices on the left hand side are the solutions of x 1 x 2 x 3 = 1, the vertices on the right hand side are the solutions of x 1 x 4 x 5 = −1.We used Remarks 3.3 and 3.4 to reduce the number of colors needed in the graph.

δ = 1 .
Next we must show that for a, b ∈ V (G), we have v a,b = 0 whenever c(a) = c(b), i.e., the colors of a and b are different.Recall from the definition of G(M, b) that the color of the vertex (l, α) is l.Thus v satisfies this relation by definition.Lastly, we must show that for a, b, a ′ , b ′ ∈ V (G), we have v a,b v a ′ ,b ′ = 0 whenever the edges {a, a ′ } and {b, b ′ } have different colors (or when one is an edge and the other is not).The previously shown relation already implies this one unless c(a) = c(b) and c(a ′ ) = c(b ′ ).So we may assume that a = (l, α), b = (l, β), a ′ = (k, α ′ ), and b ′ = (k, β ′ ).Thus, letting δ = α△β and δ δ ′ .Suppose the edges {a, a ′ } and {b, b ′ } do have different colors.By definition of G(M, b) this is equivalent to α△α ′ = β△β ′ .

αũ(k) α = 1 .
can be used to show that, for fixed k ∈ [m] the ũ(k) α are pairwise orthogonal projections satisfying α∈±1 S k 0 For i ∈ S k and β ∈ ±1 S k 0 we have that u (k)

2 . 3 ]Lemma 4 . 1 .
. The degree deg(v) of a vertex v denotes the number of neighbors of v in the graph G. Let G be a finite graph, A G be its adjacency matrix and u vw , 1 ≤ v, w ≤ n be the generators of C(Qut(G)).If (A l G ) vv = (A l G ) ww for some l ∈ N, then u vw = 0. Particularly, if deg(v) = deg(w), then u vw = 0.

1 . 4 . 4 .
since we have deg(p) = deg(q) and thus u pq = 0 by Lemma 4.Remark Lemma 4.1, Lemma 4.2 and Lemma 4.3 also work for vertex-and edge-colored graphs (take the decolored adjacency matrix in Lemma 4.1), since colors just add more relations on the generators of C(Qut(G)).

Lemma 4 . 6 .
Let G be a vertex -and edge-colored graph with deg(v) ≥ 3 for all v ∈ V (G) and let G ′ be the graph as in Definition 4.5.Denote by v i , 0 ≤ i ≤ n c(v) , the vertices in the added path with

0 for all i by Step 3 .Theorem 4 . 7 .
The case n c(v) > n c(w) follows similarly.Let G be a vertex -and edge-colored graph with deg(v) ≥ 3 for all v ∈ V (G) and let G ′ be as in Definition 4.5.Then there exists a * -isomorphism ϕ : C

Step 5 :
It holds u e i f i = 0 for c(e) = c(f ).Since c(e) = c(f ), we know m c(e) = m c(f ) .First assume m := m c(e) < m c(f ) .Then we have u emfm = 0 by Lemma 4.1, since deg(e m ) = 1 = 2 = deg(f m ).We deduce u e i f i = u e 0 f 0 = u emfm = 0 for all i by Step 3. The case m c(e) > m c(f ) is similar.

Corollary 4 . 14 .
Let G be a vertex -and edge-colored graph such that deg(v) ≥ 3 for all v ∈ V (G).Denote by G ′ and G ′′ the graphs constructed in Definitions 4.5 and 4.9.If Qut

1 .
c(g a ) = c ′ (g ′ a ) and c(g b ) = c ′ (g ′ b ); 2. g a = g b if and only if g ′ a = g ′ b ; Based on the above, we define the following 4 : for any x ∈ {+1, −1}.Therefore, k ∈ [m] such that i ∈ S k , and these operators are entries of the magic unitary u, we have that y * i = y i .Also, by (23) we have that u for α = β and thus

( 3 )
of Definition 6.5 holds, i.e., thati∈S k y i = (−1) b k +b ′ k for all k ∈ [m].We have that i∈S k ′ 0 (for the same color by assumption, for different colors because the product is 0).Note that w ′′ is a magic unitary if and only if the matrices u ′ i are magic unitaries.Let e = (a, b) ∈ E ′ 0 .We compute mwhere we put(u ′ i ) e i f i = w ′ ak w ′ bl +w ′ al w ′ bk for e = (a, b), f = (k, l).Those elements are projections by Lemma 4.8, because we have w ′ ak w ′ bl = w ′ bl w ′ ak for (a, b), (k, l) ∈ E f ;nc(f )≥i