On Wiener polarity index of bicyclic networks

Abstract

Complex networks are ubiquitous in biological, physical and social sciences. Network robustness research aims at finding a measure to quantify network robustness. A number of Wiener type indices have recently been incorporated as distance-based descriptors of complex networks. Wiener type indices are known to depend both on the network’s number of nodes and topology. The Wiener polarity index is also related to the cluster coefficient of networks. In this paper, based on some graph transformations, we determine the sharp upper bound of the Wiener polarity index among all bicyclic networks. These bounds help to understand the underlying quantitative graph measures in depth.

In order to decide whether a given network is robust, a way to quantitatively measure network robustness is needed. Intuitively robustness is all about back-up possibilities, or alternative paths, but it is a challenge to capture these concepts in a mathematical formula. During the past years a lot of robustness measures have been proposed¹. Network robustness research is carried out by scientists with different backgrounds, like mathematics, physics, computer science and biology. As a result, quite a lot of different approaches to capture the robustness properties of a network have been undertaken. All of these approached are based on the analysis of the underlying graph—consisting of a set of vertices connected by edges of a network¹,²,³,⁴,⁵,⁶.

One such category is the distance-based descriptors which include Wiener index, Harary index, etc. The use of Wiener index and related type of indices dates back to the seminal work of Wiener in 1947⁷. Wiener introduced his celebrated index to predict the physical properties, such as boiling point, heats of isomerization and differences in heats of vaporization, of isomers of paraffin by their chemical structures. Wiener index has since inspired many distance-based descriptors in Chemometrics. These include Harary index⁸, hyper Wiener index⁹,¹⁰, Wiener polynomial¹¹, Balaban index¹², Wiener polarity index⁷ and information indices¹³,¹⁴,¹⁵. These indices, or commonly called descriptors, play significant roles in quantitative structure-activity relationship/quantitative structure-property relationship (QSAR/QSPR) models. It is known that the Wiener type indices depend both on a network’s number of nodes and its topology. For more results, we refer to¹⁶,¹⁷.

Let G = (V, E) be a connected simple graph. The distance between two vertices u and v in G, denoted by d_G(u, v), is the length of a shortest path between u and v in G. The Wiener polarity index of a graph G = (V, E), denoted by W_p(G), is the number of unordered pairs of vertices {u, v} of G such that d_G(u, v) = 3, i.e.,

The name “Wiener polarity index” is introduced by Harold Wiener⁷ 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 Wiener⁷ used a liner formula of W and W_P to calculate the boiling points t_B of the paraffins, i.e., where a, b and c are constants for a given isomeric group. The Wiener index W(G) is popular in chemical literatures. For more results on Wiener index, we refer to the survey paper¹⁸ written by Dobrynin, Entringer and Gutman, and some recent papers¹⁹,²⁰,²¹,²²,²³.

The Wiener polarity index is used to demonstrate quantitative structure-property relationships in a series of acyclic and cycle-containing hydrocarbons by Lukovits and Linert²⁴. Hosoya in²⁵ found a physical-chemical interpretation of W_p(G). Du, Li and Shi²⁶ described a linear time algorithm APT for computing the Wiener polarity index of trees, and characterized the trees maximizing the Wiener polarity index among all trees of given order. From then on, the Wiener polarity index started to attract the attention of a remarkably large number of mathematicians and so many results appeared. The extremal Wiener polarity index of (chemical) trees with given different parameters (e.g. order, diameter, maximum degree, the number of pendants, etc.) were studied, see27,28,29,30,31,32,33. Moreover, the unicyclic graphs minimizing (resp. maximizing) the Wiener polarity index among all unicyclic graphs of order n were given in³⁴. There are also extremal results on some other graphs, such as fullerenes, hexagonal systems and cactus graph classes, we refer to³⁵,³⁶,³⁷. Observe that the Wiener polarity index is also related to the cluster coefficient of networks.

Results

The main contributions of this paper can be summarized as follows:

• We provide a formula of the Wiener polarity index of bicyclic networks, from which the value of the index can be computed easily.

• We introduce three graph transformations, which can be used to increase the values of Wiener polarity index. These transformations can help to find more extremal values for other classes of molecular networks.

• We determine the maximum value of the Wiener polarity index of bicyclic networks and characterize the corresponding extremal graphs.

Now let us introduce some notations. Let N_G(v) be the neighborhood of v, and denote the degree of vertex v. For , we call the ith neighborhood of v. If d_G(v) = 1, then we call v a pendant vertex of G. Let g(C_x) be the length of cycle C_x in graph G, P_i denote a path with length i. For all other notations and terminology, not given here, see e.g.38.

Let B be a bicyclic graph. Suppose and are two cycles in B with l (l ≥ 0) common vertices. Without loss of generality, we label the vertices of C_p in the clockwise direction, and the vertices of C_q in the inverse clockwise direction. If l = 0, then there is one unique path P connecting C_p and C_q, which starts with v₁ and ends with u₁. We call this kind of bicyclic graph type I (see Fig. 1). If l = 1, then C_p and C_q have exactly one common vertex v₁(u₁). We call this kind of bicyclic graphs type II (see Fig. 1). If l  ≥ 2, then B contains exactly three cycles. The third cycle is denoted by C_z, where z = p + q − 2l + 2. Without loss of generality, assume that p ≤ q ≤ z and l − 2 ≤ p − 2 ≤ q − 2. The two cycles C_p and C_q have more than one common vertex . We call this kind of bicyclic graphs type III (see Fig. 1). In the following section, we use B, C_p, C_q, v_i (1 ≤ i ≤ p), u_j (1 ≤ j ≤ q), l as defined above, except as noted.

Figure 1. The three types of bicyclic graphs.

Let be the bicyclic graph of type I, where P = v₁u₁ and . Especially, we denote this kind of graphs by, if , (i = 2, 3), . For a graph G = (V, E) and , we can construct a new graph H by identifying v₁ with , denoted by , and we say P_l is incident to vertex v.

Theorem 0.1. Let B₁ be a bicyclic graph in type I and , be the desired graph attaining the maximum Wiener polarity index.

1. If n = 6, then , and ;

2. If n = 7, then , and ;

3. If n = 8, then ,, where P₁ is incident to the pendant vertex of v₁, and ;

4. If n = 9, then ,, where the path P₁ is incident to the pendant vertex of v₁,, where the path P₁ is incident to one pendant vertex of v₁, , where the two paths P₁ are incident to the pendant vertex of v₁, and ;

5. If n = 10, then ,, where the path P₁ is incident to one pendant vertex of v₁, , where the two paths P₁ are incident to the pendant vertices of v₁, and ;

6. If n = 11, then ,, where the path P₁ is incident to one pendant vertex of v₁,, where the two paths P₁ are incident to the pendant vertices of v₁, , where the three paths P₁ are incident to the pendant vertices of v₁, and ;

7. If n = 12, then , , where P₁ is incident to one pendant vertex of v₁, , where the two paths P₁ are incident to the pendant vertices of v₁, , where the three paths P₁ are incident to the pendant vertices of v₁, and ;

8. If n = 13, then , , , where P₁ is incident to one pendent vertex of v₁, , where P₁ is incident to one pendant vertex of v₁,, where the two paths P₁ are incident to the pendant vertices of v₁, , where the two paths P₁ are incident to the pendant vertices of v₁, , where the three paths P₁ are incident to the pendant vertices of v₁,, where the three paths P₁ are incident to the pendant vertices of v₁,, where the four paths P₁ are incident to the pendant vertices of v₁, and ;

9. If n = 14, then , , , where P₁ is incident to one pendent vertex of v₁, , where the two paths P₁ are incident to the pendant vertices of v₁, , where the three paths P₁ are incident to the pendant vertices of v₁, , where the four paths P₁ are incident to the pendant vertices of v₁, and ;

10. If n ≥ 15, then , and . □

Let be the bicyclic graph in type II, where , and s₁ = t₁. When n is large enough, it can be easily checked that the graph maximizing the Wiener polarity index is (see support information).

Theorem 0.2. Let B₂ be a bicyclic graph in type II and , be the desired graph attaining the maximum Wiener polarity index.

1. If n = 5, then , and W_p(B₂) = 0;

2. If n = 6, then ,, and ;

3. If n = 7, then ,, and ;

4. If n = 8, then ,, and ;

5. For n ≥ 9, let .

If r = 0, then ,, and ;

If r = 1, then , and ;

If r = 2, then , and .□

Let be the bicyclic graph in type III, where , s₁ = t₁, s₂ = t₁ and l = 1. Let be the bicyclic graph in type III, where , s₁ = t₁, s₂ = t₁ and l = 1. When n is large enough, it can be checked that the graph maximizing the Wiener polarity index is .

Theorem 0.3. Let B₃ be a bicyclic graph in type III and, be the desired graph attaining the maximum Wiener polarity index.

1. If n = 4, then , and W_p(B₃) = 0;

2. If n = 5, then , and ;

3. If n = 6, then , where P₂ is incident to vertex v₁ or v₃, and ;

4. If n = 7, then , where the two paths P₁ are incident to the pendant vertex of v₁, and ;

5. If n = 8, then , where the three paths P₁ are incident to the pendant vertex of v₁, and ;

6. If n = 9, then , where the four paths P₁ are incident to the pendant vertex of v₁, , where the three paths P₁ are incident to the pendant vertices of v₁, and ;

7. If n = 10, then , where the four paths P₁ are incident to the pendant vertices of v₁, and ;

8. If n = 11, then , where the five paths P₁ are incident to the pendant vertices of v₁,, where the four paths P₁ are incident to the pendant vertices of v₁,,, and ;

9. For n ≥ 12, let .

If r = 0, then , , and ;

If r = 1, then ,, and ;

If r = 2, then , and . □

Theorem 0.4. Let B be a bicyclic graph of order n (≥4), B^* be the bicyclic graph with the maximum polarity index among all bicyclic graphs.

1. If n = 4, then , and W_p(B₃) = 0;

2. If n = 5, then , and ;

3. If n = 6, then , and ;

4. If n = 7, then , , where the two paths P₁ are incident to the pendant vertex of v₁ and ;;

5. If n = 8, then , , where P₁ is incident to one pendant vertex of v₁, , where the three paths P₁ are incident to the pendant vertex of v₁, and ;

6. If n = 9, then , , where the path P₁ is incident to the pendant vertex of v₁, , where the path P₁ is incident to one pendant vertex of v₁, , where the two paths P₁ are incident to the pendant vertex of v₁, , , where the four paths P₁ are incident to the pendant vertex of v₁, , where the three paths P₁ are incident to the pendant vertices of v₁, and ;

7. If n = 10, then , , where the path P₁ is incident to one pendant vertex of v₁, , where the two paths P₁ are incident to the pendant vertices of v₁, , , where the four paths P₁ are incident to the pendant vertices of v₁, and ;

8. For n ≥ 11, let .

If r = 0, then , , and ;

If r = 1, then , and ;

If r = 2, then , and . □

Discussion

Quantifying the structure of complex networks is still intricate because the structural interpretation of quantitative network measures and their interrelations have not yet been explored extensively. In this paper, we studied sharp upper bounds for the Wiener polarity index among all bicyclic networks, by using some transformations. The graphs attaining these bounds are also characterized. The proof techniques use structural properties of the graphs under consideration and it may be intricate to extend the techniques when using more general graphs.

An interesting thing is that the Wiener polarity index is related to a pure mathematical problem: counting the number of subgraphs of a graph. This counting problem is a basic problem in mathematics but much more complicated. For example, Alon and Bollobás provide some results on this topic, e.g.39,40,41.

As a future work, we will consider the extremal problems of the Wiener polarity index for general networks and also some special networks. Furthermore, we would like to explore advanced structural properties of the Wiener polarity index, and relations between the Wiener polarity index and some other topological indices. On the other hand, it would be interesting to investigate the applications of Wiener polarity index in characterizing the structure properties of complex networks and studying algorithm theory and computational complexity. For instance, one can consider the possibility of using the Wiener polarity index or other distance measures to study other very interesting algorithms, like the google algorithm in complex networks⁴²,⁴³.

Methods

First we introduce some operations on bicyclic graphs, then we give the corresponding lemmas which state that the Wiener polarity index is not decreasing after applying these operations on bicyclic graphs.

Let B be a bicyclic graph. As we have claimed, suppose and are two cycles. If both and are stars, then we denote such a bicyclic graph by, where s_i and t_j represent the number of pendant vertices of v_i and u_j, respectively.

We define Operation I as follows. Let T_B[v] denote a hanging tree on vertex v of a bicyclic graph B with p ≥ 4, q ≥ 4, where v is on the cycle of B. Among all hanging trees, suppose is one of the longest paths from the root v to a leaf c_r in T_B[v]. If r ≥ 2, then after deleting the edge vc₁ from B, we obtain a bicyclic graph A and a tree T such that and . Let B^* denote the bicyclic graph obtained from A and T by identifying c₁ and v′ (a neighbor of v on the cycle of B) and adding a new hanging leaf vx to v.

We define Operation II as follows. Let B be a bicyclic graph with p = 3, T_B[v_i] be a hanging tree rooted at v_i (i = 1, 2, 3). Let be one of the longest paths from the root v_i to a leaf c_r of the hanging tree T_B[v_i].

For r ≥ 3, we define a new graph B^* as follows:

For r = 2, the operation differs on the three types of bicyclic graphs.

(1) For bicyclic graphs in type I, we let

where is on the path .

(2) For bicyclic graphs in type II, by considering the value of q, there are two cases.

Case 1. q ≥ 4. In this case, let

where is the root vertex mentioned above.

Case 2. q = 3 and . Here we let C_q = v₁v₄v₅v₁. We define an operation as follows: delete T_B[v_i]\v_i and add a copy of T_B[v_i] to v_j by identifying v_j and which is a copy of v_i. We call this operation “move T_B[v_i] to v_j”. By considering the number of vertices on the cycles of B with hanging trees, there are two subcases.

Subcase 2.1. There is only one vertex with a hanging tree. Let , .

For the case v_i = v₁, we apply operations as follows. If b ≥ 4, then move , to v₂ and (3 ≤ j ≤ b) to v₃; if b = 3, then move , to v₂ and , to v₃; if b = 2, then move , to v₂ and to v₃; if b = 1, then move to v₂ and to v₃. The new graph is denoted by B^*.

For the case v_i = v₂, we construct a new graph .

Subcase 2.2. There are at least two vertices v_s, v_t with hanging trees. In this subcase, let , where .

(3) For the bicyclic graphs in type III. By considering the value of q, there are two cases.

Case 1. q ≥ 4. In this case, we can apply Operation 1 on C_z.

Case 2. q = 3 and . Here let C_q = v₁v₂v₄v₁. We can move T_B[v₄] to v₃ to get a new graph B′ satisfying W_p(B′) = W_p(B). By considering the number of vertices on the cycles of B′ with hanging trees, there are two subcases.

Subcase 2.1. There exists only one vertex, say , which has a hanging tree. Firstly, move T_B′[v_i] to v₃ (denote the new graph by B″), delete a vertex in and meanwhile subdivide edge v₁v₄ (denote the new graph by ); secondly, move all the other vertices in to v₁ (denote the new graph by B″′); thirdly, if , then just move one pendant vertex of v₁ to v₂; if , then move one pendant vertex of v₃ to v₂.

Subcase 2.2. There exist two vertices, say , which have hanging trees. If i = 1 and j = 2, then move T_B′[v₂] to v₃. Now we can only consider the case i = 1 and j = 3.

If there exists , then delete c₂ and subdivide the edge v₁v₄ (denote the new graph by B″). Now return to the situation in Case 1.

If and , then delete a vertex and subdivide the edge v₁v₄. Now return to Case 1. For the situation that , delete a vertex and subdivide the edge v₁v₄, move all pendant vertices in to v₂, at last move one pendant vertex of v₁ or v₂ to v₃.

Subcase 2.3. There exist three vertices which have hanging trees. By deleting some pendant vertex in , where , and meanwhile subdividing the edge v₁v₄, we return to the situation in Case 1.

The final graph obtained after the above operation is denoted by B^*.

We define Operation III as follows. Let B be a bicyclic graph. If d_B(v) = 2, then let , where v′, , . We call such an operation smooth v to x.

We define Operation IV as follows. Let B be a bicyclic graph, where and are both stars. Denote the set of the pendant vertices of v_i(u_j) by V_i(U_j).

For bicyclic graphs in type I, we will take the following two steps.

Step 1. For C_p and , if i is odd, then move V_i to v₁ and smooth v_i to v₂; if i is even, then move V_i to v₂ and smooth v_i to v₁. For C_q and , if j is odd, then move U_j to u₁ and smooth u_j to u₂; if j is even, then move U_j to u₂ and smooth u_j to u₁. Therefore, we obtain a graph with a unique path P connecting C_p and C_q. Let the set of hanging leaves of u₁, u₂, u_q be , , , respectively.

Step 2. Let , (1 ≤ k ≤ t).

If k is odd, then move W_k to v₃ and smooth w_k to v₂; if k is even, then move W_k to v₂ and smooth w_k to v₃.

If t is odd, then move to v₂, to v₁, to v₃; if t is even and t ≥ 2, then move to v₃, to v₁, and to v₂, respectively; if t = 0, and d(v₂) = d(v₃) = 2, let and , then for the situation that b = 1, move b₁ to v₂ and move a₁ to v₃, for the situation that b ≥ 2, move b₁ to v₂ and move to v₃; if t = 0 and d(v_i) = 2, , then move to v_i, to v₁, to v_j, respectively.

Finally, we get a new graph and there is a unique path P = v₁u₁ connecting C_p and C_q.

For bicyclic graphs in type II, we also give two steps as follows.

Step 1. For C_p and , if i is odd, then move V_i to v₁ and smooth v_i to v₂; if i is even, then move V_i to v₂ and smooth v_i to v₁. For C_q and , if j is odd, then move U_j to u₁ and smooth u_j to u₂; if j is even, then move U_j to u₂ and smooth u_j to u₁. Thus we get a graph with s₁ = t₁. Let the set of hanging leaves of u₁, u₂, u_q be , , , respectively.

Step 2. By moving to v₂, to v_p, we have with s₁ = t₁.

For bicyclic graphs in type III, the operation is defined as follows. Recall that we use l (≥1) to denote the number of common vertices of C_p and C_q, and without loss of generality, assume l − 2 ≤ p − 2 ≤ q − 2.

1. If p ≥ 3 and q ≥ 4, then we will take the following three steps. Step 1. For , . If i is odd, then move V_i to v₁; if i is even, then move V_i to v₂; if j is odd, then move U_j to v₁; if j is even, then move U_j to v₂; move U_q to v_p. Step 2. If l = 2 or 3, smooth vertices to v₁ and v₂ alternately. If l ≥ 4, then we first smooth vertices to v₁ and v₂ alternately; then smooth vertices to v₁ and v₂ alternately. After applying this operation, we get a new graph B′ with cycles C_p′, C_q′ and C_z′. Let l′ be the number of common vertices of C_p′ and C_q′, p′ (p′ = 3 or 4) be the number of vertices of the smallest cycle of B′, then we have l′ = 2 or l′ = 3. Now relabel the vertices on C_p′ and C_q′ of B′, and we have and . Step 3. Considering the value of l′, there are two cases. Case 1. l′ = 2. We just smooth to v₁ and v₂ alternately, and smooth u_q′−1 to v_p′. The new graph obtained is denoted by . Case 2. l′ = 3, and with s₁ = t₁. Let . If q′ ≥ 5, then smooth v₃ to vertex v₁, smooth to v₁ and v₂ alternately and smooth u_q′−1 to v₄; if q′ = 4 and , then we do nothing; if q′ = 4 and , then move the pendant vertices of v₂ to v₄. Finally, we get the desired graph .

2. If p = 3 and q = 3, by Operation II on B and its resultant graphs repeatedly, we have a new graph . Move the pendant vertices of u₃ to v₃, we obtain .

Additional Information

How to cite this article: Ma, J. et al. On Wiener polarity index of bicyclic networks. Sci. Rep. 6, 19066; doi: 10.1038/srep19066 (2016).