Abstract
The Wiener polarity index of a graph is the number of unordered pairs of vertices of such that the distance between and is 3. Cycle-block graph is a connected graph in which every block is a cycle. In this paper, we determine the maximum and minimum Wiener polarity index of cycle-block graphs and describe their extremal graphs; the extremal graphs of 4-uniform cactus with respect to Wiener polarity index are also discussed.
1. Introduction
Let be a connected simple graph. The distance between the vertices and of is defined as the length of a shortest path connecting and . is called the th neighbor set of . is called the degree of . If , then is called a pendant vertex of . denotes a cycle of order . The girth of , denoted by , is the length of the shortest cycle of .
A block of the graph is a maximal 2-connected subgraph of . A cactus graph is a connected graph in which no edge lies in more than one cycle, such that each block of a cactus graph is either an edge or a cycle. If all blocks of a cactus are cycles, the graph is defined as cycle-block graph. In this paper, suppose the cycle-block graph consist of cycles, the length of the cycles may be different. If all blocks of a cactus are cycles of the same length , the cactus is -uniform. A hexagonal cactus is a -uniform cactus that every block of the graph is a hexagon. A vertex shared by two or more hexagons is called a cut-vertex. If each hexagon of a hexagonal cactus has at most two cut-vertices and each cut-vertex is shared by exactly two hexagons, we say that is a chain hexagonal cactus (see Figure 1(a)). A star cactus is a cactus consisting of cycles, spliced together in a single vertex (see Figure 1(b)). A star hexagonal cactus is a star cactus in which every cycle is a hexagon.
(a)
(b)
The Wiener polarity index of , denoted by , is the number of unordered vertex pairs of distance 3. It was first used in a linear formula to calculate the boiling points of paraffin [1]: where are constants for a given isomeric group.
The Wiener polarity index became popular recently, and many mathematical properties and its chemical applications were discovered [2–6]. In this line, Du et al. [2] characterized the minimum and maximum Wiener polarity index among all trees of order , and Deng [3] determined the largest Wiener polarity indices among all chemical trees of order . M. H. Liu and B. L. Liu [4] determined the first two smallest Wiener polarity indices among all unicyclic graphs of order . Hou et al. [5] determined the maximum Wiener polarity index of unicyclic graphs. Behmarama et al. [6] computed the Wiener polarity index of hexagonal cacti. To know more about cactus graph one can research [7, 8].
In this paper, we discuss the extremal graphs of Wiener polarity index of cycle-block graphs with and 4-uniform cactus.
2. The Extremal Graphs of Wiener Polarity Index of Cycle-Block Graphs with
In this section, we characterize the maximum and minimum Wiener polarity index of the cycle-block graphs with .
Suppose that and are two connected graphs. The graph obtained by identifying a chosen vertex of and another of is called the coalescence of and , denoted by . The vertex identifying and is called the coalescence vertex. The cycle-block graph which consist of cycles can be seen as the coalescence of cycles by times, .
Lemma 1. Let and be two connected graphs; suppose that is the coalescence vertex of . Then .
Proof. Suppose that and . Then, By the definition of Wiener polarity index, the result holds.
Theorem 2. Let be a cycle-block graph with and let be consisting of cycles . Then the equality holds if and only if , where is a star cactus graph.
Proof. Let be a cycle-block graph with . Here we apply induction to .
When , the graph is determined uniquely. By Lemma 1 and elementary computation, we have .
When , there are three cycles , , and . The graph is obtained from by attaching a new cycle to . There are three ways to attach the cycle . Here we suppose that is the coalescence vertex of and is the new coalescence vertex of .
By Lemma 1 and elementary computation, we have
where implies that is the star cactus graph. From the result of the three cases, it holds for the result.
Now suppose that the assertion holds for . Next we prove that the result holds for .
Suppose that is the coalescence vertex of and is the coalescence vertex of . Three cases occur
where ; it implies that is the star cactus graph. Obviously, the star cactus graph has the maximum Wiener polarity index of cycle-block graphs with . This completes the proof.
Let be a graph set that consists of the cycle-block graphs of cycles with cut-vertices and the distance between any cut-vertices being greater than 1. By a similar method with Theorem 2, we have Theorem 3; the proof is in the Appendix.
Theorem 3. Let be a cycle-block graph with and let be consisting of cycles . Then the equality holds if and only if .
Remark 4. By Theorems 2 and 3, we know that the cycle-block graph with maximum Wiener polarity index is determined uniquely while the cycle-block graph with minimum Wiener polarity index is not.
For some types of hexagonal cacti which represent common chemical structures, as an extension, we obtain the -uniform cactus with the maximum and minimum Wiener polarity index.
Corollary 5. Let be -uniform cactus with hexagons. Then ; the first equality holds if and only if there are different cut-vertices in and distance between any two vertices is greater than 1; the second equality holds if and only if is a star hexagonal cactus graph.
3. The Extremal Graphs of Wiener Polarity Index of 4-Uniform Cactus
The case of -uniform cactus’ extremal graphs is different from the cycle-block graphs with and more complex. Let be a graph obtained from a quadrilateral by attaching , , , and quadrilaterals to vertices , , , and , respectively, where (see Figure 2).
Theorem 6. Suppose that is 4-uniform with quadrilaterals. Then the equality holds if and only if , where .
Proof. We apply induction to .
When , is unique. By Lemma 1, we have ; the result follows.
When , we have
the result holds.
Suppose that the result holds for . When , there is a new quadrilateral to be attached to , where . Without loss of generality, suppose that ; there are two cases to be discussed concerning of the variable parameters of .
Case 1 . It is a star quadrilateral cactus. Three new graphs , , and are obtained by three different ways to attach a quadrilateral to .
By elementary computation and Lemma 1, we have
For these three subcases, the result follows.
Case 2 . There are also three subcases to attach a new quadrilateral to .
Subcase 1. or .
We only prove the case . can be proved similarly. By elementary computation and Lemma 1, we have
Subcase 2. or .
When , by elementary computation and Lemma 1, we have
Similarly, .
Subcase 3. The new quadrilateral attach to one of the or quadrilaterals and create a new cut-vertex different from and .
When a new quadrilateral is attached to one of the quadrilaterals. The distance between the new cut-vertex and its nearest cut-vertex is 1. By elementary computation and Lemma 1, we have
Analogously, when the new quadrilateral is attached to one of the quadrilaterals, we have .
When the distance between the new cut-vertex and its nearest cut-vertex is 2. By elementary computation and Lemma 1, we have
By the discussion of Case 2 above, for , it is easy to verify that . This completes the proof.
With the analogous method with Theorem 3, we can deduce the minimum Wiener polarity index of 4-uniform (see the proof of the Appendix).
Theorem 7. Let be 4-uniform with quadrilaterals. Then the equality holds if and only if , where is a chain quadrilateral cactus with the distance between the cut-vertices being at least 2 (see Figure 3).
Remark 8. From Theorems 6 and 7, we can conclude that the extremal 4-uniform with the maximum Wiener polarity index is not unique, but the minimum case is unique and different from the case of cycle-block graphs with . For the case of extremal cycle-block with of Wiener polarity index is very complex, we do not discuss it here.
Appendix
The proof of Theorem 3.
Proof. We proof the result with the similar method of Theorem 2.
When , the graph is . By Lemma 1 and elementary computation, we have .
When , from the process of the proof of Theorem 2, Case 3 has the minimum Wiener polarity index. It holds for the result.
Suppose that holds for the result; the smallest Wiener polarity index of cycle-block graphs is the cycle-block graph with cut-vertex and any two cut-vertices’ distance being greater than 1. Next we demonstrate that the result holds for . There are four cases to be discussed. Suppose that are cut-vertices between the cycles . is the cut-vertex of . Consider the following:
From the discussion for , we find that the result follows. By induction, it follows that the cycle-block graph has the minimum Wiener polarity index.
The proof of Theorem 7.
Proof. We prove the result with the same method of Theorem 3.
When and , from the process of the proof of Theorem 6, it holds for the result.
Suppose that holds for the result; next we prove the case of quadrilaterals. When , it is the graph that by attaching a new quadrilateral to the chain quadrilateral cactus of quadrilaterals which the distance between the cut-vertices is at least 2. There are four cases to be discussed. Suppose that are cut-vertices of the chain quadrilateral cactus with quadrilaterals. is the new coalescence vertex of . Consider the following:
When , graph is the chain quadrilateral cactus of quadrilaterals with the distance between the cut-vertices being at least 2; the result follows. By induction, it follows that the chain quadrilateral cactus with the distance between the cut-vertices being at least 2 has the minimum Wiener polarity index. This completes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the referees for the helpful comments and useful suggestions on this paper. This work is supported by National Natural Science Foundation of China (nos. 11071088, 11201156).