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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.
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 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 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}.
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 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}).
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
N(Aut(G)) = {σ ∈ S_{V} : σ Aut(G)σ^{−1} = Aut(G)}
and is the largest subgroup in S_{V} 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 ∈ 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 3.2Let 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 3.4σ ∈ 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}).
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,α_{1},α_{2}, α_{3}}
where—when expressed in Cayley cycle notation—i = (1)(2)(3)(4), α_{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.
symmetry-04-00219-t001_Table 1
The Cayley table for Aut(G).
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
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 ViergruppeV 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 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}.
symmetry-04-00219-t002_Table 2
The Cayley table for N(Aut(G)).
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
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).
symmetry-04-00219-t003_Table 3
The N(Aut(G)) images of E.
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}
To see that Aut(G) is the automorphism group for each graph relabeled by σ ∈ {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 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
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.
ρ_{G} ≡ | 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 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.
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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