# Hidden Symmetries in Simple Graphs

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

^{c}of G is the graph with vertex set V and edge set E

^{c}= {{u, v}, u, v ∈ V: {u, v}∉ E}. Graph G

_{1}= (V

_{1}, E

_{1}) is isomorphic to graph G

_{2}= (V

_{2}, E

_{2}) if there is a bijection φ : V

_{1}→ V

_{2}such that {u, v}∈ E

_{1}if and only if {φ(u), φ(v)}∈ E

_{2}. Thus, a graph isomorphism preserves adjacency. The bijection φ is the isomorphism between G

_{1}and G

_{2}and the associated graph isomorphism is denoted φ : G

_{1}→ G

_{2}.

_{V}of all permutations of V, denoted Aut(G) ⊂ S

_{V}. Furthermore, Aut(G) = Aut(G

^{c}) and if G

_{1}and G

_{2}are isomorphic graphs, then Aut(G

_{1}) is isomorphic to Aut(G

_{2}), denoted Aut(G

_{1}) ~ Aut(G

_{2}).

^{m}= e, where e is the identity element in X. If X ⊂ Y and yXy

^{−1}= X for every y ∈ Y, then X is a normal subgroup of Y, denoted X ⊲ Y. Here y

^{−1}∈ Y is the inverse of y. The normalizer N(Aut(G)) of Aut(G) in S

_{V}is the group defined by

_{V}: σ Aut(G)σ

^{−1}= Aut(G)}

_{V}for which Aut(G) ⊲ N(Aut(G)).

## 3. Hidden Symmetries of G

**Lemma 3.1**Let G = (V, E) be a simple vertex labeled graph. If σ ∈ S

_{V}and G

_{σ}is the graph obtained by relabeling the vertices of G as prescribed by σ, then σ : G → G

_{σ}is an isomorphism.

_{σ}= {{σ(i), σ(j)}: {i, j} ∈ E}. Now let V

_{σ}= V, define G

_{σ}= (V

_{σ}, E

_{σ}), and observe that σ : V → V

_{σ}is a bijection with the property that {i, j} ∈ E if and only if {σ(i), σ(j)} ∈ E

_{σ}. Thus, σ : G → G

_{σ}is an isomorphism.

**Lemma 3.2**Let G = (V, E) be a simple vertex labeled graph, σ ∈ S

_{V}, and G

_{σ}= (V

_{σ}, E

_{σ}) be the graph obtained by the σ relabeling of G’s vertices. If α ∈ Aut(G), then σασ

^{−1}∈ Aut(G

_{σ}).

_{σ}is an isomorphism (Lemma 3.1), then so is σ

^{−1}: G

_{σ}→ G and diagram (2) commutes, where “⇢” denotes that the diagram is completed by the map β = σασ

^{−1}. But β is an isomorphism because it is a composition of the isomorphisms σ, α, and σ

^{−1}. Therefore, β = σασ

^{−1}∈ Aut(G

_{σ}) since it is the isomorphism β : G

_{σ}→ G

_{σ}.

**Theorem 3.3 (Hidden Permutation Symmetries)**Let G = (V, E) be a simple vertex labeled graph and G

_{σ}be the graph obtained by the σ relabeling of G’s vertices. If σ ∈ N(Aut(G)), then σ : G → G

_{σ}is an isomorphism for which Aut(G

_{σ}) = Aut(G).

_{σ}is an isomorphism is established by Lemma 3.1. Recall from Lemma 3.2 that—since σ ∈ N(Aut(G)) ⊂ S

_{V}—for each α ∈ Aut(G) there is a distinct β = σασ

^{−1}∈ Aut(G

_{σ}). However, because σ ∈ N(Aut(G)), then by definition (1) it is also the case that β ∈ Aut(G) so that Aut(G

_{σ}) ⊆ Aut(G). Furthermore, β ∈ Aut(G) implies β = σασ

^{−1}for some α ∈ Aut(G) and σ ∈ N(Aut(G)). Consequently, β ∈ Aut(G

_{σ}) so that Aut(G) ⊆ Aut(G

_{σ}). Thus, Aut(G

_{σ}) = Aut(G).

_{σ}) ~ Aut(G) when σ ∈ S

_{V}. However, when σ ∈ N(Aut(G)) the group isomorphism is the identity map.

**Corollary 3.4**σ ∈ N(Aut(G)) is a hidden permutation symmetry for G

^{c}.

^{c}) = Aut(G), then it must be the case that N(Aut(G

^{c})) = N(Aut(G)) so that σ ∈ N(Aut(G)) if and only if σ ∈ N(Aut(G

^{c})). It follows from Theorem 3.3 that σ : G

^{c}→ G

^{c}

_{σ}is an isomorphism for which Aut(G

^{c}

_{σ}) = Aut(G

^{c}).

## 4. Example: Hidden Symmetries of a Simple Vertex Labeled (4, 5) Graph

#### 4.1. The Automorphism and Normalizer Groups for G

_{1},α

_{2}, α

_{3}}

_{1}= (13)(2)(4), α

_{2}= (24)(1)(3), and α

_{3}= (13)(24) (here, i is clearly the group identity element). The Cayley table for Aut(G) is easily determined from these and is given by Table 1.

i | α_{1} | α_{2} | α_{3} | |
---|---|---|---|---|

i | i | α_{1} | α_{2} | α_{3} |

α_{1} | α_{1} | i | α_{3} | α_{2} |

α_{2} | α_{2} | α_{3} | i | α_{1} |

α_{3} | α_{3} | α_{2} | α_{1} | i |

**Z**

_{4}and the Viergruppe

**V**of Felix Klein. Inspection of Table 1 reveals that Aut(G) ≁

**Z**

_{4}because there is no fourth order element in Aut(G). Thus, it must be the case that Aut(G)∼

**V**(this is further corroborated from the table by the facts that Aut(G) is an abelian group and that every Aut(G) element is order two—which are properties of

**V**).

_{V}. Trial and error yields

_{1}, α

_{2}, α

_{3}, σ

_{1}, σ

_{2}, σ

_{3}, σ

_{4}}

_{1}= (1234), σ

_{2}= (1432), σ

_{3}= (14)(23), and σ

_{4}= (12)(34). The Cayley table for N(Aut(G)) is presented as Table 2. As an aside—observe from Table 2 that N(Aut(G)) is a nonabelian group. Consequently, N(Aut(G)) must be isomorphic to either the quaternion group

**Q**or the dihedral group

**D**

_{4}since these are the only nonabelian groups of order eight. It is also seen from a closer examination of Table 2 that N(Aut(G)) is generated by σ

_{1}and α

_{1}which satisfy the relations (σ

_{1})

^{4}= i, (α

_{1})

^{2}= i, and α

_{1}σ

_{1}α

_{1}= σ

_{2}= σ

_{1}

^{−1}. Since these are precisely the generators and relations that define

**D**

_{4}then it must be the case that N(Aut(G)) ∼

**D**

_{4}.

i | α_{1} | α_{2} | α_{3} | σ_{1} | σ_{2} | σ_{3} | σ_{4} | |
---|---|---|---|---|---|---|---|---|

i | i | α_{1} | α_{2} | α_{3} | σ_{1} | σ_{2} | σ_{3} | σ_{4} |

α_{1} | α_{1} | i | α_{3} | α_{2} | σ_{3} | σ_{4} | σ_{1} | σ_{2} |

α_{2} | α_{2} | α_{3} | i | α_{1} | σ_{4} | σ_{3} | σ_{2} | σ_{1} |

α_{3} | α_{3} | α_{2} | α_{1} | i | σ_{2} | σ_{1} | σ_{4} | σ_{3} |

σ_{1} | σ_{1} | σ_{4} | σ_{3} | σ_{2} | α_{3} | i | α_{1} | α_{2} |

σ_{2} | σ_{2} | σ_{3} | σ_{4} | σ_{1} | i | α_{3} | α_{2} | α_{1} |

σ_{3} | σ_{3} | σ_{2} | σ_{1} | σ_{4} | α_{2} | α_{1} | i | α_{3} |

σ_{4} | σ_{4} | σ_{1} | σ_{2} | σ_{3} | α_{1} | α_{2} | α_{3} | i |

#### 4.2. The Hidden Permutation Symmetries of G

_{1}, α

_{2}, and α

_{3}either fix vertex labels 2 and 4 or permutes them, whereas σ

_{1}, σ

_{2}, σ

_{3}, and σ

_{4}relabel 2 and 4 as 1 and 3, or vice versa. Thus—as automorphisms—i, α

_{1}, α

_{2}, and α

_{3}must preserve adjacency by mapping edge {2,4} in G to edge {2,4} in the associated relabeled graphs and—as isomorphisms—σ

_{1}, σ

_{2}, σ

_{3}, and σ

_{4}must preserve adjacency by mapping edge {2,4} in G to edge {1,3} in the associated relabeled graphs. This is evidenced in Table 3 which lists the N(Aut(G)) image of each edge in G in the associated relabeled graph. There the bold face first column lists the edges in G and the bold face first row lists the elements of N(Aut(G)). The table entries are the N(Aut(G)) images of G edges in the corresponding relabeled graphs. For example, the image of edge {2,3} in G under the map α

_{3}is the edge {1,4} in the graph with vertices relabeled by α

_{3}. It is obvious from this table that σ : G → G

_{σ}, σ ∈ N(Aut(G)), is an isomorphism because {i,j} ∈ E if and only if {σ(i),σ(j)} ∈ E

_{σ}(i.e., σ : V → V

_{σ}= V is an edge preserving bijection).

i | α_{1} | α_{2} | α_{3} | σ_{1} | σ_{2} | σ_{3} | σ_{4} | |
---|---|---|---|---|---|---|---|---|

{1,2} | {1,2} | {2,3} | {1,4} | {3,4} | {2,3} | {1,4} | {3,4} | {1,2} |

{2,3} | {2,3} | {1,2} | {3,4} | {1,4} | {3,4} | {1,2} | {2,3} | {1,4} |

{3,4} | {3,4} | {1,4} | {2,3} | {1,2} | {1,4} | {2,3} | {1,2} | {3,4} |

{1,4} | {1,4} | {3,4} | {1,2} | {2,3} | {1,2} | {3,4} | {1,4} | {2,3} |

{2,4} | {2,4} | {2,4} | {2,4} | {2,4} | {1,3} | {1,3} | {1,3} | {1,3} |

_{1}, σ

_{2}, σ

_{3}, σ

_{4}} = N(Aut(G))

**-**Aut(G) (i.e., that each such isomorphism σ : G → G

_{σ}is Aut(G) invariant), observe that the automorphisms of Aut(G) are the only bijective vertex maps which preserve adjacency in each G

_{σ}and map edge {1,3} in each G

_{σ}to itself. For example, the set of edges in the graph relabeled by σ

_{2}(the sixth column in Table 3) is bijectively mapped in an adjacency preserving manner onto itself by α

_{2}∈ Aut(G) according to the mappings given by (3) (the associated vertex maps appear in parentheses). Similar results also hold for i, α

_{1}, and α

_{3}so that Aut(G) is the automorphism group for this σ

_{2}relabeled graph, i.e., Aut(G) is invariant under the isomorphism σ

_{2}.

## 5. Closing Remarks

_{σ}of G with an automorphism group that is isomorphic to that of G, only those permutations in the normalizer of G’s automorphism group yield G

_{σ}’s with automorphism groups identical to that of G. These special permutations define automorphism group invariant isomorphisms of G—i.e., they are the hidden (permutation) symmetries of G. Thus, each hidden permutation symmetry of G specifies a way in which G can be relabeled without changing its underlying fundamental (obvious) symmetry.

_{G}= | Aut(G)|. An analogous extension of this to a measure which includes the hidden permutation symmetries in G that are not in Aut(G) is the RUSC number.

_{G}≡ | N(Aut(G)) |

_{G}= ρ

_{G}− α

_{G}and the ratio η

_{G}≡ δ

_{G}/ρ

_{G}also provide additional measures of a system’s RUSC.

_{G}= 4, ρ

_{G}= 8, δ

_{G}= 4, and η

_{G}= ½. Thus, there are 8 system configurations which have identical fundamental symmetries. Four of these reconfigurations are defined by permutations in the set N(Aut(G))

**-**Aut(G) and comprise half of the total number of possible reconfigurations.

## Acknowledgments

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Parks, A.D.
Hidden Symmetries in Simple Graphs. *Symmetry* **2012**, *4*, 219-224.
https://doi.org/10.3390/sym4010219

**AMA Style**

Parks AD.
Hidden Symmetries in Simple Graphs. *Symmetry*. 2012; 4(1):219-224.
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**Chicago/Turabian Style**

Parks, Allen D.
2012. "Hidden Symmetries in Simple Graphs" *Symmetry* 4, no. 1: 219-224.
https://doi.org/10.3390/sym4010219