Elaborar el marco teórico referencial que permita el desarrollo de un modelo

The number of distinct isomorphism classes of networks is the number of distinct orbits ofSnon

the set of all possible adjacency matrices Xnr (and Xn≤r for inhomogeneous regular networks).

If we write OrbXnr(Sn) (or OrbXn_≤r(Sn)) for the set of orbits, we want to compute|OrbXnr(Sn)|

(or|OrbXn≤r(Sn)|).

2.4.1 Preliminaries

Let G be a group and H be a subgroup of G. For each g ∈ G, the coset gH is defined to be the set

{gx:x∈H}

Thus gH is the set of all those elements of G obtained by combining the fixed element g with the elements of H in turn.

The number of distinct cosets ofH inGis called theindexof H inG, written|G:H|. The distinct cosets ofH partitionGinto|G:H|sets each containing|H|elements. Therefore, we have

|G:H| × |H|=|G| (2.3) Lemma 2.4. If H is a subgroup of the group G, andg1, g2 ∈G, then

g1H =g2H⇔g−1₂ g1 ∈H

Proof. See Slomson (1991).

We call the set of group elements ofGwhich fix a particularx∈X, thestabilizerofx, and we denote it by Sx. Thus

Sx ={g∈G:g·x=x}

Lemma 2.5. If a groupGacts on a set X, then for each x∈X, the setSx is a subgroup ofG.

Proof. See Slomson (1991).

There is a very close relationship between the size of the orbit and the size of the stabilizer: Theorem 2.1 (Orbit-Stabilizer Theorem). Let G be a group which acts on a setX. Then

for each x∈X,

|Orbx| × |Sx|=|G| (2.4)

Proof. We can show that there is a one-to-one correspondence between the elements of Orbx

The Orbit-Stabilizer Theorem relates the number of elements in each orbit to the number of elements in each stabilizer, but as the next theorem shows, it can easily be rearranged to give a formula for the number of distinct orbits of the setX (not for each element ofX).

Theorem 2.2. Let G be a finite group which acts on a set X. Then the number of distinct

orbitsk is given by k= 1 |G| X x∈X |Sx|

Proof. This is well known but we include it for completeness.

Suppose that there arekdistinct orbits Orbx1, . . . ,Orbxk. For anytsuch that 1≤t≤k,

X x∈Orb_xt |Sx| = X x∈Orb_xt |G| |Orbx| = |G| |Orbxt|

× |Orbxt|, since Orbx= Orbxt for each x∈Orbxt

= |G|

Since the sum of the numbers|Sx|taken over the elements of just one orbit comes to|G|in each

case, when we take the sum over all the elements ofX we get a total of |G|for each orbit and hencek|G|in total. That is,

X x∈X |Sx| = k X t=1 X x∈Orb_xt |Sx| = k X t=1 |G| = k|G| Therefore, k= 1 |G| X x∈X |Sx|

This theorem enable us to calculate the number of distinct orbits easily if |X| is not too large. For example, assume that we want to calculate the number of topologically distinct three-cell homogeneous regular networks of valency 1. There are|X31|= 3

3 _{= 27 elements in}

the set of all possible adjacency matrices for three-cell homogeneous networks of valency 1 and the group acts on this set is S3. We count group elements which fix each element x ∈ X31;

however, this calculation is not practical if|X|is too large.

Notice that what we are counting is the same as counting the number of elementsx∈X31

depending on the problem, while the groupG acting on the setX remains the same. Thus the problem of evaluating the sum becomes much easier by replacing the sum overXby a sum over

G. Therefore, instead of the number of elements

{g∈G:g·xi=xi}

we use the number of elements in the set

{x∈X:gi·x=x}

This is the set of those elements of X which are fixed by the group element gi. We call it the

fixed-point set of gi, written Fix(gi). The sum over G is thus Pg∈G|Fix(g)|. Thus we can

replace the sumP

x∈X|Sx|by

P

g∈G|Fix(g)|and deduce the following theorem called theOrbit

Counting Theoremby Aldosray and Stewart (2005).

Theorem 2.3 (Orbit Counting Theorem). Let a finite groupG act on a finite setΩ. Then

|OrbΩ(G)|= 1 |G| X g∈G |FixΩ(g)| where FixΩ(g) ={x∈Ω :g·x=x}

is the fixed-point set of g.

Proof. If g, h∈Gare conjugate then g=khk−1 for somek∈G. Hence, forx∈FixΩ(g)

g·x=x ⇔ (khk−1)·x=x

⇔ k·h·(k−1·x) =x

⇔ h·(k−1·x) =k−1·x

Therefore, k−1·xis fixed by h. Thus,

FixΩ(h) =k−1·x ⇔ kFixΩ(h) =x

⇔ kFixΩ(h) = FixΩ(g)

Since the fixed-point set of g is equal to the fixed-point set of h multiplied by somek∈G, we have |F ixΩ(g)|= |F ixΩ(h)|. Therefore, we may compute one fixed-point space per conjugacy

class and weight the sum by the size of the conjugacy class. That is, if the conjugacy classes are

C1, C2, . . . , Ct and we pick a unique elementgi ∈Ci for eachi, then

|OrgΩ(G)|= 1 |G| t X i=1 |Ci||FixΩ(gi)| (2.5)