Topological Indicators

Topological indicators (or indices, also known as connectivity indices) are commonly used in (chemical) graph theory and some have been applied to fullerene theory,[41]e.g., the Wiener index, hyper-Wiener index, Balaban index, Szeged index, Estrada index, mean topological distance, and many more. A topological indicator is a graph invariant natural or real number, i.e., it does not depend on the2D representation of a graph G. It is how- ever not required that a topological index uniquely defines a graph up to isomorphism, only the converse is true. In the following we introduce the topological indices which are implemented in our program.

The EstradaE(G)index and bipartivity β(G)index are obtained from the eigenvalues of the adjacency matrixA(G),{x_i,i=1, ...,nv} ∈[−3, 3], of

a fullerene graphG,[145,270] E(G) = nv

∑

i=1 exi _and β(_G) = nv

∑

i=1 cosh(x_i)/E(G) (2.10) The WienerW(G)and hyper-WienerWW(G)indices are defined as,[130]

W(G) =1 2_{u,v}⊆

∑

_V(G)d(u,v) and (2.11) WW(G) =1 2_{u,v}⊆

∑

_V(G) d(u,v) +d(u,v)2 (2.12) Here V(G) is the vertex set of a graph G and d(u,v) is the topological (chemical) distance between the two vertices uand v, i.e., the number of edges in the shortest path connecting them. If we define

dmax_i =max

j {Dij} (2.13)

we can define the topological diametersDand radiusRas[139]

D=max{dmax_i } and R=min{dmax_i } (2.14) We can now define the reverse Wiener index[271]_as

The Balaban index is defined as[30,142]

B(G) = ne

ne−nv+2_i<j

∑

_∈E(G)

W_iW_j−1/2, (2.16) whereneare the number of edges andWiis the sum of topological distances

between vertexiand all other vertices in the graph,

Wi =

∑

j∈V(G)

d(i,j) (2.17)

For fullerenes with equal row sums in the matrix d(i,j), i.e., min(W_i) = max(W_i), the following relationship between the Wiener and Balaban indices holds,[139]

W(G)B(G) = 9

4

n3_v

nv+4 (2.18)

The Szeged index is defined as[272]

Sz(G) =

∑

e∈E(G)

n_ij(e)n_ji(e) (2.19) wheren_ij(e) =|B_ij(e)|is the number of vertices that are closer to vertexi

than vertexjin an edgee= (ij)∈E(G)(withE(G)being the edge set) and

B_ij(e) ={k|k∈V(G),D_ik<D_jk} (2.20) Further mentioned here are two topological efficiency parametersρandρE

defined by Vukicevic, Ori and co-workers,[135,273]

ρ=2 W(G)

nvWmin with Wmin=min{Wi} (2.21)

and

ρE= Wmax

W_min with Wmax=max{Wi} (2.22)

The spectral moments{μk,k=0, 1, . . .}are calculated from eigenvaluesxi

of the adjacency matrix,

μk = nv

∑

i=1

and we have μ0 = nv, μ1 = 0, μ2 = 3nv, μ3 = 0, μ4 = 15nv, μ5 = 120,

μ4=93nv−120, andμ7=1680. The higher spectral moments depend on

the fullerene structure.[274]

One of the first topological indicators introduced to discuss the stability of fullerenes are the neighbor indices for pentagons and hexagons. The program calculates the pentagon neighbor indices,[32,275]hexagon neighbor indices and the pentagon arm indices.[276]Every fullerene isomer can be characterized by a signature of the form {ik|k= 1, . . . ,n}. The pentagon

indices(p_i|i=0, . . . , 5)define the number of pentagons attached to another pentagon, i.e. forIPRfullerenes p0=12 and all other pentagon indices are

zero. Hexagon indices are similarly defined, i.e.(h_i|i=0, . . . , 5), whereh_k

is the number of hexagons with neighbor indexk. In anIPRfullerene every hexagon is adjacent to a minimum of three others and we can restrict the list to(h3,h4,h5,h6). The pentagon arm indices(ni|i=0, . . . , 5)counts the

number of arms of the pentagons. Here an edgeEincident to a vertex of a pentagon not belonging to the pentagon is called an arm if, i) both end- vertices of edgeEare incident to pentagons, and ii)Eshares two neighbor hexagons.[276]This is easily seen to be a Stone-Wales pattern.

We can now define some useful topological invariants.[32]_{The pentagon}

indexNpis defined as Np= 1 2 5

∑

k=1 kpk with 5

∑

k=0 pk=12 (2.24)

ForIPRfullerenes we have no pentagon attached to another and therefore

p0=12 and all otherpi =0, and it follows thatNp=0. ForIPRfullerenes

a more useful index is defined through the hexagon neighbor indices. The standard deviationσhof the hexagon neighbor index distribution is defined

as σh= k2 _{− k}2 _(2.25) where kn= ∑ 6 k=1knhk ∑6 k=0hk with 6

∑

k=0 hk= n 2 −10 (2.26) Réti and László define the pentagon arm indexNA in a similar way to the

pentagon index,[276] NA = 1 2 5

∑

k=1 kn_k with 5

∑

k=0 n_k=12 (2.27)

The first and second moments M₁and M2and the variance Var are defined

as (real numbers), M_i= 1 12 5

∑

k=1 kin_k and Var=M2−M12 (2.28)

We now define the Réti-László topological descriptor asΨ,

Ψ= 30+6M1

1+4.5Np+C(M1,M2) with (2.29) C(M1,M2) = (120M2)

1/2

(1+7M₁)1/2_(1+0.9Var1.5₎ (2.30)

If there exists a non-negative integer qfor which one of the arm indices

nq = 12, the fullerene is called q-balanced. In this case Var = 0.[276] If Np+NA =0, the fullerene is called strongly isolated. C60 contains Stone-

Wales patterns and is therefore not strongly isolated. The first leapfrog of C60, C180, and the ones to follow are strongly isolated.

2.11 Volume, Surface Area and Deviation from

In document Graph theoretic and electronic properties of fullerenes, &, Biasing molecular modelling simulations with experimental residual dipolar couplings : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy at Masse (Page 86-89)