On Wiener polarity index of cactus graphs††thanks: Supported by National Natural Science Foundation of China (11071002, 11126178), Program for New Century Excellent Talents in University, Key Project of Chinese Ministry of Education (210091), Specialized Research Fund for the Doctoral Program of Higher Education (20103401110002), Science and Technological Fund of Anhui Province for Outstanding Youth (10040606Y33), Project of Educational Department of Anhui Province (KJ2011A019), Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University(KJJQ1001), Academic Innovation Team of Anhui University Project (KJTD001B).
Abstract: The Wiener polarity index of a graph is the number of unordered pairs of vertices such that the distance between and is . In this paper we give an explicit formula for the Wiener polarity index of cactus graphs. We also deduce formulas for some special cactus graphs.
MR Subject Classifications: 05C12, 92E10
Keywords: Wiener polarity index; distance; cactus graph
We use Trinajstić  for terminology and notation. Let be a connected graph. The distance between two vertices and in , denoted by , is the length of a shortest path between and in . The Wiener polarity index of a graph , denoted by , is defined by
which is the number of unordered pairs of vertices of such that . In organic compounds, say paraffin, this number is the number of pairs of carbon atoms which are separated by three carbon-carbon bonds. The name “Wiener polarity index” for the quantity defined in (1) is introduced by Harold Wiener  in 1947. Wiener himself conceived the index only for acyclic molecules and defined it in a slightly different-yet equivalent-manner. In the same paper, Wiener also introduced another index for acyclic molecules, called Wiener index or Wiener distance index and defined by
Wiener  used a linear formula of and to calculate the boiling points of the paraffins, i.e.,
where and are constants for a given isomeric group.
The Wiener index is popular in chemical literatures. In the mathematical literature, it seems to be studied firstly by Entringer et al.  in 1976. From then on, many researchers studied the Wiener index in different ways. For instance, one can see , , , , , , ,  and  for the theoretical aspects, and ,  and  for algorithmic and computational aspects. Recently, Dobrynin et al. gave a comprehensive survey  for the Wiener index. The reader is referred to the paper for further details.
In the best of our knowledge, Wiener had some information about the applicability of this topological index. Using the Wiener polarity index, Lukovits and Linert  demonstrated quantitative structure property relationships in a series of acyclic and cycle-containing hydrocarbons. Hosoya  found a physico-chemical interpretation of . Recently, Du et al.  described a linear time algorithm for computing the Wiener polarity index of trees and characterized the trees maximizing the index among all the trees of the given order. Deng et al.  characterized the extremal trees with respect to this index among all trees of order and diameter . Deng  also gave the extremal Wiener polarity index of all chemical trees with order . Deng and Xiao  found the maximum Wiener polarity index of chemical trees with vertices and pendants.
However, it seems that less attention has paid for Wiener polarity index of cycle-containing graph up to now. While we are preparing this paper, we find that Behmaram et al.  discuss Wiener polarity index of fullerenes and hexagonal systems which contain no triangles or quadrangles, and Hou et al.  discuss the maximum Wiener polarity index of unicyclic graphs. In the paper we consider the Wiener polarity index of cactus graphs which are allowed to have triangles or quadrangles or many cycles.
2 Main result
In this section, we introduce some graphs used in this paper. Firstly, we introduce two graphs and as follows; see Fig. 2.1.
Fig. 2.1 The graphs and
Let be a graph. Denote by the number of cycles of with length . The numbers of the induced subgraph of and in are denoted by and , respectively.
A cactus graph is a connected graph in which no edge lies in more than one cycle. A -gon cactus graph is a cactus graph in which every block is a -gon or , where denotes a cycle of length . If each -gon of a -gon cactus has at most two cut-vertices, and each cut-vertex is shared by exactly two hexagons, then is called a chain -gon cactus. If, in addition, any two cut-vertices on a -gon has distance (respectively, at least ), then this chain -gon cactus is said of type 1 (respectively, type 2); see Fig. 2.2. For a -gon cactus of type 1 (respectively, type 2), expanding each of the cut-vertices to an edge, we will get a graph called ortho-chain -gon cactus (respectively, meta-chain -gon cactus); see Fig. 2.3.
Fig. 2.2 Chain hexagonal cactuses of type 1 (left side) and type 2 (right side)
Fig. 2.3 Ortho-chain hexagonal cactus (left side) and meta-chain hexagonal cactus(right side)
Let be a connected cactus graph. Then
Proof. We consider the edge of the graph and choose a vertex adjacent to and another vertex adjacent to . Then we introduce a three-edge set in , denoted by , defined by
One can easily get .
If , namely coincide and in a common , then .
If , namely lie in a common , then .
If , namely lie in a common , or lie in a common while does not, or lie in a common while does not, then .
If , then may lie in a common , or may lie in a common while does not, or may lie in a common while does not, then
Combining the above discussion, we have
Let be a connected cactus graph such that every triangle or quadrangle has exactly one neighbor. Then
Proof. Put and in Theorem 2.1.
Let be a chain -gon cactus of type 1 with -gons. Then
Proof. It is easy to get
So, by Theorem 2.1, if , then
and if , then
Similarly, we get the remaining result and omit the details.
Let be a chain -gon cactus of type 2 with -gons (). Then
Proof. One can get
So, if , . The remaining proof is omitted.
Let be an ortho-chain -gon cactus with -gons. Then
Proof. One can get
So, if , ; and if , . The remaining proof is omitted.
Let be a meta-chain -gon cactus with -gons. Then
Proof. One can get
So, if , . The remaining proof is omitted.
-  C.A. Barefootć, R.C. Entringerć and L.A. Székelyć, Extremal values for rations of distances in trees, Discrete Math., 80(1997), 37-56.
-  A. Behmarama, H. Yousefi-Azaria, A. R. Ashrafi, Wiener polarity index of fullerenes and hexagonal systems, Applied Math. Lett., 25(10)(2012), 1510-1513.
-  D. Bonchevć and O.E. Polanskyć, The Wiener number of graphs, I. General theory and change due to graph operations, MATH Commun. Math. Comput. Chem., 21 (1986), 133-186.
-  D. Bonchev and N. Trinajstić, Information theory, distance matrix and molecular branching, J. Chem. Phys., 67(1977), 4517-4533.
-  E. R. Canfield, R. W. Robinson and D. W. Rouvray, Determination of the Wiener molecular branching index for the general tree, J. Comput. Chem., 6(1985), 598-609.
-  P. Dankelmann, R.C. Entringer, Average distance, minimum degree, and spanning trees, J. Graph Theory, 33(1)(2000), 1-13.
-  H. Deng, H. Xiao, F. Tang, On the extremal Wiener polarity index of trees with a given dimeter, MATH Commun. Math. Comput. Chem., 63(2010), 257-264.
-  H. Deng, On the extremal Wiener polarity index of chemical trees, MATH Commun. Math. Comput. Chem., 66(2011), 305-314.
-  H. Deng, H. Xiao, The Wiener polarity index of molecular graphs of alkanes with a given unmber of methyl groups, J. Serd. Chem. Soc., 75(2010), 1405-1412.
-  A. A. Dobrynin, R. C. Entringer and I. Gutman, Wiener index of trees:theory and applications, Acta Appl. Math., 66(2001), 211-249.
-  W. Du, X. Li, Y. Shi, Algorithms and extremal problem on Wiener polarity index, MACH Commun. Math. Comput. Chem., 62(2009), 235-244.
-  R. C. Entringer, A. Meir, J. W. Moon and L. A. Székely, On the Wiener index of trees from certain families, Australas. J. Combin., 10(1994), 211-224.
-  R. C. Entringer, D. E. Jackson and D. A. Snyder, Distance in graphs, Czechoslovak Math. J., 26(1976), 283-296.
-  I. Gutman, A new method for the calculation of the Wiener number of acyclic molecules, J. Mol. Struct. (Theochem), 285(1993), 137-142.
-  I. Gutman, S. L. Lee, Y. L. Luo and Y. N. Yeh, some recent results in the theory of Wiener index, Indian J. Chem., 32A(1993), 137-142.
-  H. Hou, B. Liu, Y. Huang, The maximum Wiener polarity index of unicyclic graphs, Applied Math. Comput., 218(20)(2012), 10149-10157.
-  H. Hosoya, Mathematical and chemical analysis of Wiener’s polarity number, in:Topology in Chemistry Discrete Mathematics of Molecules (D. H. Rouvaray, R. B. King Eds.), Horwood, Chichester, 2002.
-  I. Lukovits, W. Linert, Polarity-numbers of cycle-containing structures, J. Chem. Inf. Comput. Sci., 38(1998), 715-719.
-  B. Mohar, Eigenvalues, diameter and mean distance in graphs, Graph Combin., 7(1991), 53-64.
-  B. Mohar and T. Pisanski, How to compute the Wiener index of a graph, J. Math. Chem., 2(1988), 267-277.
-  N. Trinajstić, Chemical graph theory, CRC Press, Boca Raton, FL, 1993.
-  H. Wiener, Strutural determination of paraffin boiling points, J. Amer. Chem. Soc., 69(1947), 17-20.