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Symmetry 2012, 4(1), 219-224; https://doi.org/10.3390/sym4010219

Article
Hidden Symmetries in Simple Graphs
Electromagnetic and Sensor Systems Department, 18444 Frontage Road Suite 327, Naval Surface Warfare Center Dahlgren Division, Dahlgren, VA 22448-5161, USA
Received: 15 February 2012; in revised form: 23 February 2012 / Accepted: 27 February 2012 / Published: 5 March 2012

Abstract

:
It is shown 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 permutation of Vi.e., σ is a hidden permutation symmetry of G. A simple example illustrates the theory and the applied notion of system robustness for reconfiguration under symmetry constraint (RUSC) is introduced.
Keywords:
graph theory; automorphism group; normalizer; hidden symmetry; symmetry measures

1. Introduction

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.

2. Preliminaries

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 Gc of G is the graph with vertex set V and edge set Ec = {{u, v}, u, vV: {u, v}∉ E}. Graph G1 = (V1, E1) is isomorphic to graph G2 = (V2, E2) if there is a bijection φ : V1V2 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 : G1G2.
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 SV of all permutations of V, denoted Aut(G) ⊂ SV. Furthermore, Aut(G) = Aut(Gc) 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 xX is the least positive integer m such that xm = e, where e is the identity element in X. If XY and yXy−1 = X for every yY, then X is a normal subgroup of Y, denoted XY. Here y−1Y is the inverse of y. The normalizer N(Aut(G)) of Aut(G) in SV is the group defined by
N(Aut(G)) = {σSV : σ Aut(G)σ −1 = Aut(G)}
and is the largest subgroup in SV for which Aut(G) ⊲ N(Aut(G)).

3. Hidden Symmetries of 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 ∈ SV and Gσ is the graph obtained by relabeling the vertices of G as prescribed by σ, then σ : G → Gσ 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 Eσ = {{σ(i), σ(j)}: {i, j} ∈ E}. Now let Vσ = V, define Gσ = (Vσ, Eσ), and observe that σ : V → Vσ is a bijection with the property that {i, j}E if and only if {σ(i), σ(j)} ∈ Eσ. Thus, σ : G → Gσ is an isomorphism.
Lemma 3.2 Let G = (V, E) be a simple vertex labeled graph, σ ∈ SV, and Gσ = (Vσ, Eσ) be the graph obtained by the σ relabeling of G’s vertices. If α ∈ Aut(G), then σασ −1∈ Aut(Gσ).
Proof. Since σ : G → Gσ is an isomorphism (Lemma 3.1), then so is σ −1: Gσ → 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(Gσ) since it is the isomorphism β : Gσ → Gσ.
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Theorem 3.3 (Hidden Permutation Symmetries) Let G = (V, E) be a simple vertex labeled graph and Gσ be the graph obtained by the σ relabeling of G’s vertices. If σ ∈ N(Aut(G)), then σ : G → Gσ is an isomorphism for which Aut(Gσ) = Aut(G).
Proof. The fact that σ : G → Gσ is an isomorphism is established by Lemma 3.1. Recall from Lemma 3.2 that—since σ ∈ N(Aut(G)) ⊂ SV—for each α ∈ Aut(G) there is a distinct β = σασ −1 ∈ Aut(Gσ). However, because σ ∈ N(Aut(G)), then by definition (1) it is also the case that β ∈ Aut(G) so that Aut(Gσ) ⊆ Aut(G). Furthermore, β ∈ Aut(G) implies β = σασ −1 for some α ∈ Aut(G) and σ ∈ N(Aut(G)). Consequently, β ∈ Aut(Gσ) so that Aut(G) ⊆ Aut(Gσ). Thus, Aut(Gσ) = Aut(G).
Note that in general Aut(Gσ) ~ Aut(G) when σSV. However, when σN(Aut(G)) the group isomorphism is the identity map.
Corollary 3.4 σ ∈ N(Aut(G)) is a hidden permutation symmetry for Gc.
Proof. Since Aut(Gc) = Aut(G), then it must be the case that N(Aut(Gc)) = N(Aut(G)) so that σ ∈ N(Aut(G)) if and only if σ ∈ N(Aut(Gc)). It follows from Theorem 3.3 that σ : Gc→ Gcσ is an isomorphism for which Aut(Gcσ) = Aut(Gc).

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

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

4.1. The Automorphism and Normalizer Groups for G

By inspection it is found that
Aut(G) = {i,α12, α3}
wherewhen expressed in Cayley cycle notation—i = (1)(2)(3)(4), α1 = (13)(2)(4), α2 = (24)(1)(3), and α3 = (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.
Table 1. The Cayley table for Aut(G).
Table 1. The Cayley table for Aut(G).
iα1α2α3
iiα1α2α3
α1α1iα3α2
α2α2α3iα1
α3α3α2α1i
It is interesting to note that up to (group) isomorphism there are only two groups of order four—the cyclic group Z4 and the Viergruppe V of Felix Klein. Inspection of Table 1 reveals that Aut(G)Z4 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).
In order to find N(Aut(G)) it is necessary to apply definition Equation (1) to the elements of SV. Trial and error yields
N(Aut(G)) = {i, α1, α2, α3, σ1, σ2, σ3, σ4}
where σ1 = (1234), σ2 = (1432), σ3 = (14)(23), and σ4 = (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 D4 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 σ1 and α1 which satisfy the relations 1)4 = i, (α1)2 = i, and α1σ1α1 = σ2 = σ1 −1. Since these are precisely the generators and relations that define D4 then it must be the case that N(Aut(G))D4.
Table 2. The Cayley table for N(Aut(G)).
Table 2. The Cayley table for N(Aut(G)).
iα1α2α3σ1σ2σ3σ4
iiα1α2α3σ1σ2σ3σ4
α1α1iα3α2σ3σ4σ1σ2
α2α2α3iα1σ4σ3σ2σ1
α3α3α2α1iσ2σ1σ4σ3
σ1σ1σ4σ3σ2α3iα1α2
σ2σ2σ3σ4σ1iα3α2α1
σ3σ3σ2σ1σ4α2α1iα3
σ4σ4σ1σ2σ3α1α2α3i

4.2. The Hidden Permutation Symmetries of G

In order to illustrate Theorem 3.3, first note that i, α1, α2, and α3 either fix vertex labels 2 and 4 or permutes them, whereas σ1, σ2, σ3, and σ4 relabel 2 and 4 as 1 and 3, or vice versa. Thus—as automorphisms—i, α1, α2, and α3 must preserve adjacency by mapping edge {2,4} in G to edge {2,4} in the associated relabeled graphs and—as isomorphisms—σ1, σ2, σ3, and σ4 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 α3 is the edge {1,4} in the graph with vertices relabeled by α3. It is obvious from this table that σ : G → Gσ, σ ∈ N(Aut(G)), is an isomorphism because {i,j} ∈ E if and only if {σ(i),σ(j)} ∈ Eσ (i.e., σ : V → Vσ = V is an edge preserving bijection).
Table 3. The N(Aut(G)) images of E.
Table 3. The N(Aut(G)) images of E.
iα1α2α3σ1σ2σ3σ4
{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}
To see that Aut(G) is the automorphism group for each graph relabeled by σ ∈1, σ2, σ3, σ4} = N(Aut(G)) - Aut(G) (i.e., that each such isomorphism σ : G → Gσ is Aut(G) invariant), observe that the automorphisms of Aut(G) are the only bijective vertex maps which preserve adjacency in each Gσ and map edge {1,3} in each Gσ to itself. For example, the set of edges in the graph relabeled by σ2 (the sixth column in Table 3) is bijectively mapped in an adjacency preserving manner onto itself by α2 ∈ Aut(G) according to the mappings given by (3) (the associated vertex maps appear in parentheses). Similar results also hold for i, α1, and α3 so that Aut(G) is the automorphism group for this σ2 relabeled graph, i.e., Aut(G) is invariant under the isomorphism σ2.
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5. Closing Remarks

Although every permutation relabeling σ of the vertex labels of a simple graph G defines an isomorphic copy Gσ 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 Gσ’s with automorphism groups identical to that of G. These special permutations define automorphism group invariant isomorphisms of Gi.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 αG = | 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.
ρG ≡ | N(Aut(G)) |
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 δG = ρGαG and the ratio ηGδGG also provide additional measures of a system’s RUSC.
For a system represented by the above (4,5) graph, αG = 4, ρG = 8, δG = 4, and ηG = 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

This work was supported by a grant from the Naval Surface Warfare Center Dahlgren Division’s In-house Laboratory Independent Research Program.

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