Hidden Symmetries in Simple Graphs

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 V—i.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.


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 OPEN ACCESS 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.

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 G c of G is the graph with vertex set V and edge set 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 : 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 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)).

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 ∈ S V and G σ is the graph obtained by relabeling the vertices of G as prescribed by σ, then
Proof.The relabeling of G's vertices is specified by the permutation σ : V → V so that the associated relabeled edges are the set 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, Proof.The fact that σ : However, because σ ∈ N(Aut(G)), then by definition (1) it is also the case that 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.

Table 1. The Cayley table for Aut(G)
.
It is interesting to note that up to (group) isomorphism there are only two groups of order four-the cyclic group 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).
In order to find N(Aut(G)) it is necessary to apply definition Equation (1) to the elements of S V .Trial and error yields where σ 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 Since these are precisely the generators and relations that define D 4 then it must be the case that N(Aut(G)) ∼ D 4 .

The Hidden Permutation Symmetries of G
In order to illustrate Theorem 3.3, first note that i, α 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).

Closing Remarks
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.

)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).

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 ).
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.

Table 3 .
The N(Aut(G)) images of E. Aut(G) is the automorphism group for each graph relabeled byσ ∈ {σ 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 Table3) 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 .