Cut method and Djokovic-Winkler’s relation

M. J. Nadjafi-Arani 1

Faculty of Mathematical SciencesGolpayegan University of Technology

Golpayegan, I. R. Iran

Sandi Klavzar 2

Faculty of Mathematics and PhysicsUniversity of Ljubljana

SI-1000 Ljubljana, Slovenia

Abstract

Let Θ∗ be the transitive closure of the Djokovic-Winkler’s relation Θ. It is provedthat the Wiener index of a weighted graph (G,w) can be expressed as the sum of theWiener indices of weighted quotient graphs with respect to an arbitrary combinationof Θ∗-classes. A related result for edge-weighted graphs is also given and a class ofgraphs studied in [8] is characterized as partial cubes. We will Compute distancepolynomial functions on graphs with transitive Djokovic-Winkler’s relation.

Keywords: Wiener index, weighted graph, Djokovic-Winkler’s relation, partialcube, Isometric embedding.

1 Email: mjnajafiarani@gmail.com2 Email: sandi.klavzar@fmf.uni-lj.si

Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 45 (2014) 153–157

1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

http://dx.doi.org/10.1016/j.endm.2013.11.029

1 Introduction

The cut method (see the survey [5]) turned out to be utmost handy when deal-ing with distance-based graph invariants which are in turn among the centralconcepts of chemical graph theory. The method was initiated in [6] where itwas shown how cuts can be used to compute the Wiener index (alias aver-age distance) of graphs which admit isometric embeddings into hypercubes.These graphs are know as partial cubes. About ten years later, the result wasextended in [4] to general graphs by establishing a connection between theWiener index of a graph and its canonical metric representation. (The resultof [6] is then obtained by specializing to bipartite graphs.)

In this paper, at the first, we motivate the recent paper [8] in which it isdemonstrated that the cut method is applicable also to the edge-Wiener indexand the edge-Szeged index. The results in [8] are stated for graphs that admitcertain edge partitions. we show that these graphs are precisely partial cubes,a class of graphs extensively studied by now. Later is a generalization of theabove mentioned theorem from [4]. The generalization is two-fold. First, thevariety of factor graphs is extended by allowing arbitrary combinations of theedge classes from the canonical metric representation. Second, the result isextended to weighted graphs.

A partial cube is a graph isometrically embeddable into a hypercube.Partial cubes can be characterized as the bipartite graphs with transitiveDjokovic-Winkler’s relation Θ. A more general class of graphs is obtained byomitting the requirement for a graph to be bipartite. As proved by Winkler [7],these graphs are precisely the graphs that admit isometric embeddings intothe Cartesian product of triangles. At the end of paper, we show that the cutmethod can be extended to these graphs and to any invariant Wλ(G), whereλ is a positive integer.

2 Preliminaries

We consider the usual shortest path distance and write dG(u, v) for the dis-tance in a graph G between u and v and simplify the notation to d(u, v) whenthe graph will be clear from the context. The Wiener index of G is the sumof distances between all pairs of vertices of G.

A subgraph of a graph is called isometric if the distance between anytwo vertices of the subgraph is independent of whether it is computed in thesubgraph or in the entire graph. A subgraph of a graph is called convex iffor any two vertices of the subgraph all shortest path (from the entire graph)

M.J. Nadjafi-Arani, S. Klavžar / Electronic Notes in Discrete Mathematics 45 (2014) 153–157154

between then belong to the subgraph. For a connected graph G and an edgeab of G we set Wab = {x ∈ V (G) | d(x, a) < d(x, b)}. Note that if G isbipartite then V (G) = Wab∪Wba holds for any edge ab. By abuse of languagewe consider (when appropriate) Wab also as the subgraph induced by Wab.

For a graphG, the Djokovic-Winkler’s relation Θ [1,7] is defined onE(G) asfollows. If e = xy ∈ E(G) and f = uv ∈ E(G), then eΘf if d(x, u)+ d(y, v) �=d(x, v) + d(y, u). Relation Θ is reflexive and symmetric, its transitive closureΘ∗ is hence an equivalence relation. The partition of E(G) induced by Θ∗ willbe called the Θ∗-partition.

A weighted graph (G,w) is a graph G = (V (G), E(G)) together with theweight function w : V (G) → R

+. The Wiener index W (G,w) of (G,w) isdefined as:

W (G,w) =1

2

∑u∈V (G)

∑v∈V (G)

w(u) w(v) dG(u, v) .

Clearly, if w ≡ 1 then W (G,w) = W (G).

3 Cut method and Djokovic-Winkler’s relation

As already mentioned in the introduction, the cut method was developed in [8]for the edge-Wiener/Szeged index. More precisely, the method was developedfor graphs G that admit a partition {Fi} of the edge set such that G \ Fi isa two component graphs with convex components. We close this section bypointing out that these graphs are precisely partial cubes.

Proposition 3.1 Let G be a connected graph. Then G admits a partition {Fi}of E(G) such that G \ Fi is a two component graphs with convex componentsif and only if G is a partial cube.

Let G be a connected graph and let F = {F1, . . . , Fk} be the Θ∗-partitionof G. Let E = {E1, . . . , Er} be a partition of E(G), where each set Ei is theunion of one or more Θ∗-classes. Then we say that E is a Θ∗-merging (andthat E is a refinement of F).

Lemma 3.2 Let G be a connected graph and let E = {E1, . . . , Er} be a Θ∗-merging. Then every connected component of G \ Ej, 1 ≤ j ≤ r, induces aconvex subgraph of G.

Lemma 3.3 Let G be a connected graph and let E = {E1, . . . , Er} be a Θ∗-merging. Let C and C ′ be connected components of G\Ej and let x, y ∈ V (C)

M.J. Nadjafi-Arani, S. Klavžar / Electronic Notes in Discrete Mathematics 45 (2014) 153–157 155

and x′, y′ ∈ V (C ′). If P1 and P2 are shortest x, x′- and y, y′-paths in G,respectively, then |E(P1) ∩Ej | = |E(P2) ∩Ej |.

Let G be a connected graph and let F1, . . . , Fk be a partition of E(G).Then the quotient graph G/Fi, 1 ≤ i ≤ k, is defined follows. Its vertices arethe connected components of G \ Fi, two vertices C and C ′ being adjacent ifthere exist vertices x ∈ C and y ∈ C ′ such that xy ∈ Fi.

We are now ready for the main result of this paper.

Theorem3.4 Let (G,w) be a connected ,weighted graph ,and let E={E1 , . . . ,Er}be a Θ∗-merging. Then

W (G,w) =r∑

j=1

W (G/Ej, wj) ,

where wj : V (G/Ej) → R+ is defined with wj(C) =

∑x∈C w(x), for any

connected component C of G \ Ej.

It is also natural to consider edge-weighted graphs, that is, pairs (G,wE),where G is a graph and wE : E(G) → R

+. The Wiener index W (G,wE) ofan edge-weighted graph (G,wE) is defined just as the usual Wiener index,that is, W (G,wE) =

12

∑u∈V (G)

∑v∈V (G) d(u, v), where the distance function

is of course computed in (G,wE). Again, if all the edges have weight 1, thenW (G,wE) = W (G).

With the methods parallel to those from Section 3 the following result canbe proved:

Theorem 3.5 Let (G,wE) be a connected,edge-weighted graph. If E={E1, . . . ,Er}is a Θ∗-merging such that for any j = 1, . . . , r, the edges from Ej have thesame weight, w(Ej), then

W (G,wE) =r∑

j=1

w(Ej)W (G/Ej, wj) .

Let G be a graph with transitive relation Θ and let E1, . . . , Ek be the Θ-classes. Let C

(i)1 , C

(i)2 , C

(i)3 , 1 ≤ i ≤ k, be the connected components of G−Ei,

let n(i)j = |V (C

(i)j )|, 1 ≤ j ≤ 3, and let n

i1,i2,...,ipj1,j2,...,jp

=∣∣∣C(i1)

j1∩ C

(i2)j2

∩ · · · ∩ C(ip)jp

∣∣∣and

Np =∑1≤i≤k

ji �=j′i

ni1,i2,...,ikj1,j2,...,jk

· ni1,i2,...,ipj′1,j

′2,...,j

′p.

M.J. Nadjafi-Arani, S. Klavžar / Electronic Notes in Discrete Mathematics 45 (2014) 153–157156

We are ready to Compute distance polynomial functions on graphs withtransitive Djokovic-Winkler’s relation:

Theorem 3.6 Let G be a graph with transitive relation Θ. Then for anypositive integer s,

Ws =k∑

p=1

∑tr1 ,tr2 ,...,trp>0

tr1+tr2+···+trp=s

(s

tr1, tr2 , . . . , trp

)Nrp.

References

[1] Djokovic, D., Distance preserving subgraphs of hypercubes, J. Combin. TheorySer. B 14 (1973), 263–267.

[2] Graham, R. L., and P. M. Winkler, On isometric embeddings of graphs, Trans.Amer. Math. Soc. 288 (1985), 527–536.

[3] Khalifeh, M. H., H. Yousefi-Azari, A. R. Ashrafi, and S. G. Wagner, Some newresults on distance-based graph invariants, European J. Combin. 30 (2009),1149–1163.

[4] Klavzar, S., On the canonical metric representation, average distance, andpartial Hamming graphs, European J. Combin. 27 (2006), 68–73.

[5] Klavzar, S., A bird’s eye view of the cut method and a survey of its applicationsin chemical graph theory, MATCH Commun. Math. Comput. Chem. 60 (2008),255–274.

[6] Klavzar, S., I. Gutman, and B. Mohar, Labeling of benzenoid systems whichreflects the vertex-distance relations, J. Chem. Inf. Comput. Sci. 35 (1995),590–593.

[7] Winkler, P., Isometric embeddings in products of complete graphs, DiscreteAppl. Math. 7 (1984), 221–225.

[8] H. Yousefi-Azari, M. H. Khalifeh, and A. R. Ashrafi, Calculating the edgeWiener and Szeged indices of graphs, J. Comput. Appl. Math. 235 (2011),4866–4870.

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