1 Introduction

The Edge-Wiener Index and the Edge-Hyper-Wiener Index of Phenylenes



Petra Žigert Pleteršek




Faculty of Chemistry and Chemical Engineering, University of Maribor, Slovenia

Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia

petra.zigert@um.si



(July 20, 2019)

Abstract

Besides the well known Wiener index, which sums up the distances between all the pairs of vertices, and the hyper-Wiener index, which includes also the squares of distances, the edge versions of both indices attracted a lot of attention in the recent years.

In this paper we consider the edge-Wiener index and the edge-hyper-Wiener index of phenylenes, which represent an important class of molecular graphs. For an arbitrary phenylene, four quotient trees based on the elementary cuts are defined in a similar way as it was previously done for benzenoid systems. The computation of the edge-Wiener index of the phenylene is then reduced to the calculation of the weighted Wiener indices of the corresponding quotient trees. Furthermore, a method for computing the edge-hyper-Wiener index of phenylenes is described. Finally, the application of these results gives closed formulas for the edge-Wiener index and the edge-hyper-Wiener index of linear phenylenes.

1 Introduction

The Wiener index of a graph is defined as the sum of distances between all pairs of vertices in the graph. It was introduced in 1947 by Wiener [33] and represents one of the most studied molecular descriptors. On the other hand, the hyper-Wiener index takes into account also the squares of distances and was first introduced by Randić in 1993 [30]. The research on the Wiener index, the hyper-Wiener index and other distance-based descriptors is still a popular topic, see papers [7, 13, 18, 29] for some recent investigations.

The edge-Wiener index of a graph was introduced in [21] as the Wiener index of the line graph and has been since then intensively investigated [1, 3, 6, 12, 20, 27, 28]. However, the concept was studied even before as the Wiener index of line graphs, see [4, 17]. Similarly, the edge-hyper-Wiener index was introduced in [22].

A cut method is a powerful method for efficient computation of topological indices of graphs. Some of the earliest results are related to the computation of the Wiener index of benzenoid systems and are presented in [8, 9, 25]. See also [26] for a survey paper on the cut method and [2, 10, 11, 32] for some recent investigations on this topic. In [23] a cut method for the edge-Wiener index of benzenoid systems was proposed and in [31] a cut method for the edge-hyper-Wiener index of partial cubes was developed.

Beside benzenoid hydrocarbons, phenylenes represent another interesting class of polycyclic conjugated molecules, whose properties have been extensively studied, see [14, 15]. The Wiener index and the hyper-Wiener index of phenylenes were studied in [16] and [5], respectively. In this paper, we consider methods for computing the edge-Wiener index and the edge-hyper-Wiener index of phenylenes and use them to obtain closed formulas for linear phenylenes.

2 Preliminaries

The distance between two vertices of a graph, denoted by , is defined as the length of a shortest path between and . The Wiener index of a connected graph is

To point out that it is the vertex-Wiener index, we will also write for . From some technical reasons we also set . The distance between two edges , denoted by , is the usual shortest-path distance between vertices and of the line graph of . Here we follow this convention because in this way the pair forms a metric space. Then the edge-Wiener index of a connected graph is defined as

(1)

In other words, is just the Wiener index of the line graph of . On the other hand, for edges and of a graph it is also legitimate to set

Replacing with in (1), another variant of the edge-Wiener index is obtained (see [24]) and we denote it by . However, there is an obvious connection between and :

(2)

Finally, the vertex-edge Wiener index is

where for a vertex and an edge we set

Next, we extend the above definitions to weighted graphs. Let be a connected graph and let and be given functions. Then , , and are a vertex-weighted graph, an edge-weighted graph, and a vertex-edge weighted graph, respectively. The corresponding Wiener indices of these weighted graphs are defined as

Again, we will often use for and is defined analogously as by using instead of .

For an efficient computation of the Wiener indices of weighted trees we need some additional notation. If is a tree and , then the graph consists of two components that will be denoted by and . For a vertex-edge weighted tree , , and set

We then recall the following results:

(3)
(4)
(5)

Note that Equation (3) was proved in [25], Equation  (4) in [34] and Equation (5) in [23].

The hyper-Wiener index and the edge-hyper-Wiener index of are defined as:

Let be the hexagonal (graphite) lattice and let be a cricuit on it. Then a benzenoid system is induced by the vertices and edges of , lying on and in its interior. Let be a benzenoid system. A vertex shared by three hexagons of is called an internal vertex of . A benzenoid system is said to be catacondensed if it does not possess internal vertices. Otherwise it is called pericondensed. Two distinct hexagons with a common edge are called adjacent. The inner dual of a benzenoid system is a graph which has hexagons of as vertices, two being adjacent whenever the corresponding hexagons are also adjacent. Obviously, the inner dual of a catacondensed benzenoid system is a tree.

Let be a catacondensed benzenoid system. If we add squares between all pairs of adjacent hexagons of , the obtained graph is called a phenylene. We then say that is a hexagonal squeeze of and denote it by .

Let be a phenylene and a hexagonal squeeze for . The edge set of can be naturally partitioned into sets , , and of edges of the same direction. Denote the sets of edges of corresponding to the edges in , , and by , and , respectively. Moreover, let be the set of all the edges of that do not belong to . For , set . The quotient graph , , has connected components of as vertices, two such components and being adjacent in if some edge in joins a vertex of to a vertex of . In a similar way we can define the quotient graphs of hexagonal squeeze . It is known [8] that for any benzenoid system its quotient graphs are trees. Then a tree is isomorphic to for and is isomorphic to the inner dual of .

Now we extend the quotient trees , , to weighted trees , , as follows:

  • for , let be the number of edges in the component of ;

  • for , let be the number of edges between components and .

3 A method for computing the edge-Wiener index of phenylenes

In this section we show that the edge-Wiener index of a phenylene can be computed as the sum of Wiener indices of its weighted quotient trees. The obtained result is similar as the result for the edge-Wiener index of benzenoid systems [23], but one additional quotient tree must be considered.

Theorem 3.1

Let be a phenylene. Then

Proof.  Let be a phenylene and let , , be its quotient trees. For any we define by

(6)

We will first show that

(7)

holds for any pair of edges . Let , such that . Select any shortest path from to in and define for . As is a shortest path, no two edges of belong to the same cut. Since it suffices to show that for it holds . Let be connected components of such that and . It follows that . In order to show that , we consider the following cases:

  • and .
    In this case we have and and the desired conclusion is clear.

  • Exactly one of and is in .
    We may assume without loss of generality that and . Then for some and . Since , it follows that .

  • and .
    Now for some and for some . We thus get that .

Since in all possible cases , Equation (7) holds. Applying this result we obtain

The obtained sums can be divided into three sums regarding the function from Equation (6):

Application of the definition of the weighted trees finally results in

The weighted Wiener indices of trees can be computed in linear time by using Equations (3), (4), and (5). Also, the quotient trees can be obtained in linear time as well, for the details see [23]. Consequently, we obtain the following corollary.

Corollary 3.2

Let be a phenylene with edges. Then the edge-Winer index of can be computed in time.

4 A method for computing the edge-hyper-Wiener index of phenylenes

In this section, we briefly introduce the method for computing the edge-hyper-Wiener index of phenylenes, which is based on a general method for partial cubes, see [31]. First, we state some important definitions.

Two edges and of graph are in relation , , if

Note that this relation is also known as Djoković-Winkler relation. The relation is reflexive and symmetric, but not necessarily transitive [19].

The hypercube of dimension is defined in the following way: all vertices of are presented as -tuples where for each and two vertices of are adjacent if the corresponding -tuples differ in precisely one coordinate. A subgraph of a graph is called an isometric subgraph if for each it holds . Any isometric subgraph of a hypercube is called a partial cube. For an edge of a graph , let be the set of vertices of that are closer to than to . We write for the subgraph of induced by .

The following theorem gives two basic characterizations of partial cubes:

Theorem 4.1

[19] For a connected graph , the following statements are equivalent:

  • is a partial cube.

  • is bipartite, and and are convex subgraphs of for all .

  • is bipartite and .

Furthermore, it is known that when is a partial cube and is a -class of , then has exactly two connected components, namely and , where . For more details about partial cubes see [19].

To state the method for computing the edge-hyper-Wiener index of any partial cube, we need to introduce some more notation. If is a partial cube with -classes , we denote by and the connected components of the graph , where . For any distinct set

Also, for and we define

Theorem 4.2

[31] Let be a partial cube and let be the number of its -classes. Then

The sum from Theorem 4.2 will be denoted by , i.e.

Obviously, by case of Theorem 4.1 any phenylene is a partial cube. An elementary cut of a phenylene is a line segment that starts at the center of a peripheral edge of , goes orthogonal to it and ends at the first next peripheral edge of . By we sometimes also denote the set of edges that are intersected by the corresponding elementary cut. Moreover, an elementary cut of a phenylene coincides with exactly one of its -classes.

Let and be two distinct elementary cuts (-classes) of a phenylene . Since the elementary cuts can have an intersection or not, we obtain the following two options from Figure 1.

Figure 1: Two different positions of two elementary cuts.

In the following, the contribution of the pair to is denoted as . Therefore, we get

Hence, for phenylene with elementary cuts it holds

5 Linear phenylenes

A hexagon of a phenylene is called terminal if it has a common edge with only one square of , otherwise we say that it is internal. If an internal hexagon has common edges with exactly two other squares, then it has exactly two vertices of degree two. If this two vertices are not adjacent, we say that such hexagon is linear. A phenylene is called linear if all its internal hexagons are linear. A linear phenylene with exactly hexagons will be denoted by , see Figure 2.

Figure 2: Linear phenylene .

We first compute the edge-Wiener index. Therefore, we determine the weighted quotient trees from Figure 3.

Figure 3: Weighted quotient trees for the linear phenylene .

From the obtained trees it is easy to calculate the sums of weights in corresponding connected components. The results are collected in Table 1.

,
,
Table 1: Sum of weights in the corresponding connected components of quotient trees and .

Using Table 1 we can compute the corresponding Wiener indices of (which are the same also for ):

From Table 1 we also get the corresponding Wiener indices of :

However, the computations for are trivial:

Finally, using Theorem 3.1 we conclude

and by Equation (2) it follows

Next, we consider the edge-hyper-Wiener index of linear phenylenes. We denote the elementary cuts of with , , , and as shown in Figure 4.

Figure 4: Elementary cuts of a linear phenylene .

The contributions of all the pairs of elementary cuts are presented in Table 2.

Table 2: Contributions of pairs of elementary cuts for .

The expressions from Table 2 give

Finally, by Theorem 4.2 the edge-hyper Wiener index of linear phenylene is equal to

Acknowledgment

The author acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0297).

References

  • [1] Y. Alizadeh, A. Iranmanesh, T. Došlić, M. Azari, The edge Wiener index of suspensions, bottlenecks, and thorny graphs, Glas. Mat. Ser. III 49(69) (2014) 1–12.
  • [2] M. Arockiaraj, A. J. Shalini, Extended cut method for edge Wiener, Schultz and Gutman indices with applications, MATCH Commun. Math. Comput. Chem. 76 (2016) 233–250.
  • [3] M. Azari, A. Iranmanesh, A. Tehranian, A method for calculating an edge version of the Wiener number of a graph operation, Util. Math87 (2012) 151–164.
  • [4] F. Buckley, Mean distance in line graphs, Congr. Numer. 32 (1981) 153–162.
  • [5] G. Cash, S. Klavžar, M. Petkovšek, Three methods for calculation of the hyper-Wiener index of molecular graphs, J. Chem. Inf. Comput. Sci. 42 (2002) 571–576.
  • [6] A. Chen, X. Xiong, F. Lin, Explicit relation between the Wiener index and the edge-Wiener index of the catacondensed hexagonal systems, Appl. Math. Comput. 273 (2016) 1100–1106.
  • [7] Y.-H. Chen, H. Wang, X.-D. Zhang, Properties of the hyper-Wiener index as a local function, MATCH Commun. Math. Comput. Chem. 76 (2016) 745–760.
  • [8] V. Chepoi, On distances in benzenoid systems, J. Chem. Inf. Comput. Sci36 (1996) 1169–1172.
  • [9] V. Chepoi, S. Klavžar, The Wiener index and the Szeged index of benzenoid systems in linear time, J. Chem. Inf. Comput. Sci37 (1997) 752–755.
  • [10] M. Črepnjak, N. Tratnik, The Szeged index and the Wiener index of partial cubes with applications to chemical graphs, Appl. Math. Comput. 309 (2017) 324–333.
  • [11] M. Črepnjak, N. Tratnik, The edge-Wiener index, the Szeged indices and the PI index of benzenoid systems in sub-linear time, MATCH Commun. Math. Comput. Chem. 78 (2017) 675–688.
  • [12] P. Dankelmann, I. Gutman, S. Mukwembi, H. Swart, The edge-Wiener index of a graph, Discrete Math309 (2009) 3452–3457.
  • [13] A. A. Dobrynin, Hexagonal chains with segments of equal lengths having distinct sizes and the same Wiener index, MATCH Commun. Math. Comput. Chem. 78 (2017) 121–132.
  • [14] B. Furtula, I. Gutman, Ž. Tomović, A. Vesel, I. Pesek, Wiener-type topological indices of phenylenes, Indian J. Chem. 41A (2002) 1767–1772.
  • [15] I. Gutman, A. Ashrafi, On the PI index of phenylenes and their hexagonal sqeezes, MATCH Commun. Math. Comput. Chem. 60 (2008) 135–142.
  • [16] I. Gutman, G. Dömötör, Wiener number of polyphenyls and phenylenes, Z. Naturforsch 49a (1994) 1040–1044.
  • [17] I. Gutman, L. Pavlović, More on distance of line graphs, Graph Theory Notes N. Y. 33 (1997) 14–18.
  • [18] A. Ilić, M. Ilić, On some algorithms for computing topological indices of chemical graphs, MATCH Commun. Math. Comput. Chem. 78 (2017) 665–674.
  • [19] R. Hammack, W. Imrich, S. Klavžar, Handbook of Product Graphs, Second edition, CRC Press, Taylor & Francis Group, Boca Raton, 2011.
  • [20] A. Iranmanesh, M. Azari, Edge-Wiener descriptors in chemical graph theory: a survey, Curr. Org. Chem19 (2015) 219–239.
  • [21] A. Iranmanesh, I. Gutman, O. Khormali, A. Mahmiani, The edge versions of Wiener index, MATCH Commun. Math. Comput. Chem61 (2009) 663–672.
  • [22] A. Iranmanesh, A. Soltani Kafrani, O. Khormali, A new version of hyper-Wiener index, MATCH Commun. Math. Comput. Chem. 65 (2011) 113–122.
  • [23] A. Kelenc, S. Klavžar, N. Tratnik, The edge-Wiener index of benzenoid systems in linear time, MATCH Commun. Math. Comput. Chem. 74 (2015) 521–532.
  • [24] M. H. Khalifeh, H. Yousefi Azari, A. R. Ashrafi, S. G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149–1163.
  • [25] S. Klavžar, I. Gutman, Wiener number of vertex-weighted graphs and a chemical application, Discrete Appl. Math80 (1997) 73–81.
  • [26] S. Klavžar, M. J. Nadjafi-Arani, Cut method: update on recent developments and equivalence of independent approaches, Curr. Org. Chem19 (2015) 348–358.
  • [27] M. Knor, P. Potočnik, R. Škrekovski, Relationship between the edge-Wiener index and the Gutman index of a graph, Discrete Appl. Math167 (2014) 197–201.
  • [28] M. J. Nadjafi-Arani, H. Khodashenas, A. R. Ashrafi, Relationship between edge Szeged and edge Wiener indices of graphs, Glas. Mat. Ser. III 47(67) (2012) 21–29.
  • [29] H. S. Ramane, V. V. Manjalapur, Note on the bounds on Wiener number of a graph, MATCH Commun. Math. Comput. Chem. 76 (2016) 19–22.
  • [30] M. Randić, Novel molecular descriptor for structure-property studies, Chem. Phys. Lett. 211 (1993) 478–483.
  • [31] N. Tratnik, A method for computing the edge-hyper-Wiener index of partial cubes and an algorithm for benzenoid systems, Appl. Anal. Discr. Math., to appear.
  • [32] N. Tratnik, The edge-Szeged index and the PI index of benzenoid systems in linear time, MATCH Commun. Math. Comput. Chem. 77 (2017) 393–406.
  • [33] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) 17–20.
  • [34] H. Yousefi-Azari, M. H. Khalifeh, A. R. Ashrafi, Calculating the edge Wiener and edge Szeged indices of graphs, J. Comput. Appl. Math235 (2011) 4866–4870.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
332368
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description