The concept of hidden symmetries of an object was introduced by Weyl [1]. Underlying this is the notion that if X is an H-set, where H is a symmetry group (the group of obvious symmetries) acting on X, additional hidden symmetries associated with X may correspond to elements of a larger group which also acts upon X and contains H as a subgroup. Sophisticated approaches based upon Weyl’s concept for finding hidden symmetries in physical systems have found application in solving and understanding a variety of problems of scientific interest (e.g., [2,3,4,5]), including numerous applications in computer science (see, for example, the survey [6] and the monograph [7]).

The primary objective of this paper is to show that each element σ in the normalizer of the automorphism group Aut(G) of a simple graph G with labeled vertex set V is an Aut(G) invariant isomorphism between G and the graph obtained from G by the σ relabeling of V (i.e., σ is a hidden permutation symmetry of G). The remainder of this paper is organized as follows: the relevant topics in graph theory and group theory are summarized in the next section (for additional depth and clarification the reader is invited to consult such standard texts as [8] and [9]). The hidden permutation symmetries of a simple vertex labeled graph G are identified in Section 3. A simple example is presented in Section 4 to illustrate the theory. Closing remarks comprise the final section of this paper.

A simple graph G is the pair G = (V, E), where V is a finite set of at least two vertices and the edge set E is either a set of doubleton subsets of V or the empty set ∅. If {u, v}∈ E, then u and v are adjacent in G. The order of a graph G is the cardinality |V| of V, |E| is its size, and G is a (|V|, |E|) graph. G is vertex labeled when V = {1,2,3,…,n}. A labeled graph which is relabeled by a permutation σ of it vertices is the graph where vertex i is relabeled as σ(i). The complement G^c of G is the graph with vertex set V and edge set E^c = {{u, v}, u, v ∈ V: {u, v}∉ E}. Graph G₁ = (V₁, E₁) is isomorphic to graph G₂ = (V₂, E₂) if there is a bijection φ : V₁→ V₂ such that {u, v}∈ E₁ if and only if {φ(u), φ(v)}∈ E₂. Thus, a graph isomorphism preserves adjacency. The bijection φ is the isomorphism between G₁ and G₂ and the associated graph isomorphism is denoted φ : G₁→ G₂.

An automorphism of G is an isomorphism of G with itself. The set of all automorphisms of G under the operation “composition of functions” forms the automorphism (or symmetry) group Aut(G) of G. When G is vertex labeled, then Aut(G) is a subgroup of the symmetric group S_V of all permutations of V, denoted Aut(G) ⊂ S_V. Furthermore, Aut(G) = Aut(G^c) and if G₁ and G₂ are isomorphic graphs, then Aut(G₁) is isomorphic to Aut(G₂), denoted Aut(G₁) ~ Aut(G₂).

The order of a group X is |X| and the order of x ∈ X is the least positive integer m such that x^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

and is the largest subgroup in S_V for which Aut(G) ⊲ N(Aut(G)).

The automorphisms of the symmetry group Aut(G) of G are the obvious symmetries of G. The objective of this section is to show that each σ ∈ N(Aut(G)) is a hidden permutation symmetry of G—i.e., it is an Aut(G) invariant graph isomorphism between G and the graph obtained from G by the application of σ to G’s vertex labels (thus, σ ∈ Aut(G) is both a G automorphism and a hidden permutation symmetry of G). The next two lemmas are required to prove this.

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.

Proof. The relabeling of G’s vertices is specified by the permutation σ : V → V so that the associated relabeled edges are the set E_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).

Proof. Since σ : G → G_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).

Proof. The fact that σ : G → 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).

Note that in general 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.

Proof. Since Aut(G^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).

Let G = (V, E), where V = {1,2,3,4} and E = {{1,2}, {2,3}, {3,4}, {1,4}, {2,4}}.

Although every permutation relabeling σ of the vertex labels of a simple graph G defines an isomorphic copy G_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.

Various real complex systems of recent interest—such as biochemical processes, global trading patterns, and scientific collaborations—can be modeled as simple labeled graphs. Many of these systems are surprisingly highly symmetric (i.e., they possess large numbers of obvious symmetries). Within the context of complex systems the hidden permutation symmetries of the labeled graph representing a system identify the system’s robustness for reconfiguration under symmetry constraint (RUSC), i.e., the ability to reconfigure the system without changing its fundamental symmetry.

In order to better understand symmetry and its affect on system properties, effort has been devoted in recent years to developing simple measures which quantify system symmetry in terms of the automorphism group of the system’s graph model (e.g., [10,11]). The most direct measure of (obvious) symmetry in a graph G is the quantity α_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.

This quantity counts the total number of ways G (i.e., the system) can be relabeled (i.e., reconfigured) without changing the automorphism group Aut(G) (i.e., the fundamental symmetry of the system). The difference δ_G = ρ_G − α_G and the ratio η_G ≡ δ_G/ρ_G also provide additional measures of a system’s RUSC.

For a system represented by the above (4,5) graph, α_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.