arXiv subjects

Simon Schmidt

Publications and source records attributed to Simon Schmidt.

31 records · Page 2Linked to original sources

arXiv2021

Quantum symmetry vs nonlocal symmetry

We introduce the notion of nonlocal symmetry of a graph $G$, defined as a winning quantum correlation for the $G$-automorphism game that cannot be produced classically. Recent connections between quantum group theory and quantum information show that quantum correlations for this game correspond to tracial states on $C(\text{Qut}(G))$ -- the algebra of functions on the quantum automorphism group of $G$. This allows us to also define nonlocal symmetry for any quantum permutation group. We investigate the differences and similarities between this and the notion of quantum symmetry, defined as non-commutativity of $C(\text{Qut}(G))$. Roughly speaking, quantum symmetry vs nonlocal symmetry can be viewed respectively as non-classicality of our model of reality vs non-classicality of our observation of reality. We show that quantum symmetry is necessary but not sufficient for nonlocal symmetry. In particular, we show that the complete graph on five vertices is the only connected graph on five or fewer vertices with nonlocal symmetry, despite a dozen others having quantum symmetry. In particular this shows that the quantum symmetric group on four points, $S_4^+$, does not exhibit nonlocal symmetry, answering a question from the literature. In contrast to quantum symmetry, we show that two disjoint classical automorphisms do not guarantee nonlocal symmetry. However, three disjoint automorphisms do suffice. We also give a construction of quantum permutation matrices built from a finite abelian group $Γ$ and a permutation $π$ on $|Γ|$ elements. Computational evidence suggests that for cyclic groups of increasing size almost all permutations $π$ result in nonlocal symmetry. Surprisingly, the construction never results in nonlocal symmetry when $\mathbb{Z}_2^3$ is used. We also investigate under what conditions nonlocal symmetry arises when taking unions or products of graphs.

math.QA↗

arXiv2019

Uniformly vertex-transitive graphs

We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph $D(Γ)$ of a given graph $Γ$ to show that the uniform vertex-transitivity of $Γ$ is equivalent to the existence of cliques of sufficient size in $D(Γ)$. Using this method, we find examples of graphs that are vertex-transitive but not uniformly vertex-transitive, settling a previously open question. Furthermore, we develop sufficient criteria for uniform vertex-transitivity in the situation of a graph with an imprimitive automorphism group. We classify the non-Cayley uniformly vertex-transitive graphs on less than 30 vertices outside of two complementary pairs of graphs.

math.CO↗

arXiv2019

Sinkhorn algorithm for quantum permutation groups

We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply it to the question whether a given finite graph (without multiple edges) has quantum symmetries in the sense of Banica. In order to do so, we run our Sinkhorn algorithm and check whether or not the resulting projections commute. We discuss the produced data and some questions for future research arising from it.

math.QA↗

arXiv2019

Quantum automorphisms of folded cube graphs

We show that the quantum automorphism group of the Clebsch graph is $SO_5^{-1}$. This answers a question by Banica, Bichon and Collins from 2007. More general for odd $n$, the quantum automorphism group of the folded $n$-cube graph is $SO_n^{-1}$. Furthermore, we show that if the automorphism group of a graph contains a pair of disjoint automorphisms this graph has quantum symmetry.

math.OA↗

arXiv2019

Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach

The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the algebra associated to the quantum automorphism group is commutative. We use Gröbner base computations in order to tackle this problem; the implementation uses GAP and the SINGULAR package LETTERPLACE. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

math.QA↗

arXiv2019

On the quantum symmetry of distance-transitive graphs

In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs $H(n,3)$, the Johnson graphs $J(n,2)$ and the Kneser graphs $K(n,2)$ do not have quantum symmetry. We also give a table with the quantum automorphism groups of all cubic distance-transitive graphs. Furthermore, with one graph missing, we can now decide whether or not a distance-regular graph of order $\leq 20$ has quantum symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum symmetry. On a final note, we present an example of a pair of graphs with the same intersection array (the Shrikhande graph and the $4 \times 4$ rook's graph), where one of them has quantum symmetry and the other one does not.

math.CO↗

arXiv2018

Quantum Symmetries of Graph C*-Algebras

The study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

math.OA↗

arXiv2018

The Petersen graph has no quantum symmetry

In 2007, Banica and Bichon asked whether the well-known Petersen graph has quantum symmetry. In this article, we show that the Petersen graph has no quantum symmetry, i.e. the quantum automorphism group of the Petersen graph is its usual automorphism group, the symmetric group $S_5$.

math.OA↗

arXiv2017

Twins and Vertex- Identification on Graphs

Recently, several vertex identifying notions were introduced (identifying coloring, lid-coloring, ...), these notions were inspired by identifying codes. All of them, as well as original identifying code, are based on separating two vertices according to some conditions on their closed neighborhood. Therefore, twins can not be identified. So most of known results focus on twin-free graph. Here, we show how twins can modify optimal value of vertex-identifying parameters for identifying coloring and locally identifying coloring. \textbf{Keyword :} Identifying coloring, locally identifying coloring, twins, separating.

math.CO↗

arXiv2017

A New Game Invariant of Graphs: the Game Distinguishing Number

The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A distinguishing $d$-coloring is a coloring $c:V(G)\rightarrow\{1,...,d\}$ invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing number. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph $G$ with a set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $G$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then the Gentle wins if the coloring is distinguishing and the Rascal wins otherwise. This game leads to define two new invariants for a graph $G$, which are the minimum numbers of colors needed to ensure that the Gentle has a winning strategy, depending on who starts. These invariants could be infinite, thus we start by giving sufficient conditions to have infinite game distinguishing numbers. We also show that for graphs with cyclic automorphisms group of prime odd order, both game invariants are finite. After that, we define a class of graphs, the involutive graphs, for which the game distinguishing number can be quadratically bounded above by the classical distinguishing number. The definition of this class is closely related to imprimitive actions whose blocks have size $2$. Then, we apply results on involutive graphs to compute the exact value of these invariants for hypercubes and even cycles. Finally, we study odd cycles, for which we are able to compute the exact value when their order is not prime. In the prime order case, we give an upper bound of $3$.

math.CO↗

arXiv2015

Game Distinguishing Numbers of Cartesian Products of Graphs

The distinguishing number of a graph $H$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(H)$ is the least integer $d$ such that $H$ has a $d$-distinguishing coloring. A $d$-distinguishing coloring is a coloring $c:V(H)\rightarrow\{1,\dots,d\}$ invariant only under the trivial automorphism. In this paper, we continue the study of a game variant of this parameter, recently introduced. The distinguishing game is a game with two players, Gentle and Rascal, with antagonist goals. This game is played on a graph $H$ with a fixed set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $H$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then Gentle wins if the coloring is $d$-distinguishing and Rascal wins otherwise. This game defines two new invariants, which are the minimum numbers of colors needed to ensure that Gentle has a winning strategy, depending who starts the game. The invariant could eventually be infinite. In this paper, we focus on cartesian product, a graph operation well studied in the classical case. We give sufficient conditions on the order of two connected factors $H$ and $F$ relatively prime, which ensure that one of the game distinguishing numbers of the cartesian product $H\square F$ is finite. If $H$ is a so-called involutive graph, we give an upper bound of order $D^2(H)$ for one of the game distinguishing numbers of $H\square F$. Finally, using in part the previous result, we compute the exact value of these invariants for cartesian products of relatively prime cycles. It turns out that the value is either infinite or equal to $2$, depending on the parity of the product order.

math.CO↗

arXiv2015

The 3/5-conjecture for weakly $S(K_{1,3})$-free forests

The $3/5$-conjecture for the domination game states that the game domination numbers of an isolate-free graph $G$ on $n$ vertices are bounded as follows: $γ_g(G)\leq \frac{3n}5 $ and $γ_g'(G)\leq \frac{3n+2}5 $. Recent progress have been done on the subject and the conjecture is now proved for graphs with minimum degree at least $2$. One powerful tool, introduced by Bujtás is the so-called greedy strategy for \D. In particular, using this strategy, she has proved the conjecture for isolate-free forests without leafs at distance $4$. In this paper, we improve this strategy to extend the result to the larger class of weakly $S(K_{1,3})$-free forests, where a weakly $S(K_{1,3})$-free forest $F$ is an isolate-free forest without induced $S(K_{1,3})$, whose leafs are leafs of $F$ as well.

math.CO↗

arXiv2015

On the Complexity of the Misère Version of Three Games Played on Graphs

We investigate the complexity of finding a winning strategy for the misère version of three games played on graphs : two variants of the game $\text{NimG}$, introduced by Stockmann in 2004 and the game $\text{Vertex Geography}$ on both directed and undirected graphs. We show that on general graphs those three games are $\text{PSPACE}$-Hard or Complete. For one $\text{PSPACE}$-Hard variant of $\text{NimG}$, we find an algorithm to compute an effective winning strategy in time $\mathcal{O}(\sqrt{|V(G)|}.|E(G)|)$ when $G$ is a bipartite graph.

cs.DM↗