Wiener Index: Formulas for Non-homeomorphic Graphs

More documents

Recommendations

Info

450 I. LUKOVITSW is one of the most freqnently applied graph theoretical invariants. Whas been used to explain the variation in boiling points, molar volumes, refractiveindices, heats of isomerization and heats of vaporization of alkanes. 1Later heats of formation, atomization, isomerization, and vaporization aswell as density, critical pressure, surface tension, viscosity, melting points,partition coefficients, chromotographic retention indices, and stability ofcrystal lattices of various kinds of molecules were related 2,3 to W. Besideschemistry, the concept of the Wiener number has been used in electricalengineering 4 and mathematics. 5 Quite a few graph invariants related to Whave been proposed, too. 6In addition to Eq. (1), analytical formulas have been proposed for W.Wiener derived a formula for chains 1 (i.e. for acyclic structures having twoendpoints), Entringer et al. 7 and Bonchev et al. 8 proposed formulas for simplecycles, Trinajsti} et al. obtained formulas for polymers 9 and Gutman derivedformulas for various types of acyclic graphs. 10 A formula was also derivedfor fused bicyclic structures. 11 Canfield et al. proposed a recursionmethod to obtain appropriate formulas for any acyclic structure. 12 Each ofthese formulas is valid for a given set of homeomorphic structures, polymersor stars. Two graphs are homeomorphic if both can be obtained from thesame graph by a sequence of subdivisions of lines. For example any two cyclesare homeomorphic, any two chains are homeomorphic and the star depictedin Figure 1 is homeomorphic with the structure depicted in Figure 2.In this paper, a method will be proposed that allows to obtain formulas for abroader set of structures.Figure 2. Example of a »starlike« graph containing three chains. k =2,m =3,andn= 4. The branching vertex belongs to all three chains.DERIVATION OF FORMULASExpressions »vertex« and »atom«, »chemical structure« and »graph«,»bond« and »edge«, »valence« and »degree« are synonymous words used inchemistry and graph theory, respectively, and will be used interchangeablyhereafter.Consider a »starlike« graph (Figure 2) that has a single branching vertex,and three side-chains k, m, and n, where k, m and n denote both thenumber of vertices of the side chain and the side-chains themselves. A starlikegraph (Figure 2) is homeomorphic with a star which has a branching
FORMULAS FOR NON-HOMEOMORPHIC GRAPHS 451vertex of the same degree (Figure 1). The branching atom belongs to allside-chains. Example: isobutane (Figure 1) consists of three side chains, andeach side chain contains two vertices. Instead of »side-chain«, we shall usethe expression »string« to denote a chainlike subgraph starting with an endpointor a branching vertex and ending with and endpoint or a branchingvertex. An atom between these two vertices (if any) is bivalent.The Wiener index of a starlike graph containing strings k, m and n (Figure2) may be calculated by using the following formula: 2,3W = (k 3 + m 3 + n 3 )+3(k 2 m + k 2 n + m 2 n + km 2 + kn 2 + mn 2 )––6(k 2 + m 2 + n 2 )–6(km + kn + mn) +5(k + m + n)/6. (2)For isobutane with k = m = n=2, we obtain:W = (2 3 +2 3 +2 3 ) + 3(2 2 2+2 2 2+2 2 2+22 2 +22 2 +22 2 ) – 6(2 2 +2 2 +2 2 ) – 6(22 +22 +22) + 5(2 + 2 + 2)/6 = (24 + 144 – 72 – 72 + 30)/6 = 9, thesame value that was obtained by using Eq. (1). Eq. (2) remains valid if anyof the side chains disappears, i. e. the size of the corresponding string – e.g.n – is equal to 1. Let us use Eq. (2) to obtain W for propane, then k =3,m = n = 1, and W = (3 3 +1 3 +1 3 ) + 3(3 2 1+3 2 1+1 2 1+31 2 +31 2 +11 2 ) – 6(3 2 + 1 2 + 1 2 ) – 6(31 +31 +11)+5(3+1+1)/6 = (29 + 78 – 66 – 42 + 25)/6 = 4, in accordance with thevalue that could be obtained for the (hydrogen suppressed) graph of propane.We want to derive a similar formula for a starlike graph containing fourstrings k, m, n and o (Figure 3) and the formula should have the followingform:W = A 4 (k 3 + m 3 + n 3 + o 3 )+B 4 (k 2 m + k 2 n + k 2 o + m 2 n + m 2 o + n 2 o ++km 2 + kn 2 + ko 2 + mn 2 + mo 2 + no 2 )+D 4 (k 2 + m 2 +n 2 + o 2 )++ E 4 (km + kn + ko + mn + mo+ no) +F 4 (k + m + n + o)/6 (3)Figure 3. Example of a »starlike« graph containing four chains. The branching vertexbelongs to all four chains, k =2,m =3,n =4,ando =5.
Page 1: CROATICA CHEMICA ACTA CCACAA 71 (3)
Page 7 and 8: FORMULAS FOR NON-HOMEOMORPHIC GRAPH
Page 9 and 10: FORMULAS FOR NON-HOMEOMORPHIC GRAPH

Wiener Index: Formulas for Non-homeomorphic Graphs

Create successful ePaper yourself

Delete template?

Save as template?