Wiener Polarity Index of Cycle-Block Graphs

The Wiener polarity indexW P of a graph G is the number of unordered pairs of vertices u, V of G such that the distance d G (u, V) between u and V is 3. Cycle-block graph is a connected graph in which every block is a cycle. In this paper, we determine the maximum and minimumWiener 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.


Introduction
Let  = (, ) be a connected simple graph.The distance   (, V) between the vertices  and V of  is defined as the length of a shortest path connecting  and V.    () = {V ∈ () |   (, V) = } is called the th neighbor set of .() = | 1   ()| is called the degree of .If () = 1, then  is called a pendant vertex of .  =  1  2 ⋅ ⋅ ⋅    1 denotes a cycle of order (≥ 3).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 6-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.
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]: of paraffin: 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][3][4][5][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  ≥ 5 and 4-uniform cactus.

The Extremal Graphs of Wiener Polarity
Index of Cycle-Block Graphs with  ≥ 5 In this section, we characterize the maximum and minimum Wiener polarity index of the cycle-block graphs with  ≥ 5. Suppose that  1 and  2 are two connected graphs.The graph obtained by identifying a chosen vertex of  1 and another of  2 is called the coalescence of  1 and  2 , denoted by  1 ∘  2 .The vertex identifying  1 and  2 is called the coalescence vertex.The cycle-block graph which consist of  cycles  1 ,  2 , . . .,   can be seen as the coalescence of cycles by By the definition of Wiener polarity index, the result holds.
By Lemma 1 and elementary computation, we have where (V 0 , V 1 ) = 0 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  =  − 1.Next we prove that the result holds for  = .
Suppose that V 0 is the coalescence vertex of where (V 0 , V 1 ) = 0; it implies that  is the star cactus graph.Obviously, the star cactus graph has the maximum Wiener polarity index of cycle-block graphs with  ≥ 5.This completes the proof.
Let G be a graph set that consists of the cycle-block graphs of  cycles with −1 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.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 6uniform cactus with the maximum and minimum Wiener polarity index.
Corollary 5. Let  be 6-uniform cactus with  hexagons.Then 11−8 ≤   () ≤ 4 2 −; the first equality holds if and only if there are  − 1 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.

The Extremal Graphs of Wiener Polarity Index of 4-Uniform Cactus
The case of 4-uniform cactus' extremal graphs is different from the cycle-block graphs with  ≥ 5 and more complex.
When  = 3, we have the result holds.Suppose that the result holds for  =  − 1.When  = , there is a new quadrilateral to be attached to  −1 (, , 0, 0), where  =  − 2 − ,  ≥ 0. Without loss of generality, suppose that  > 0; there are two cases to be discussed concerning of the variable parameters of .
By elementary computation and Lemma 1, we have For these three subcases, the result follows.
When  ≅   (, , 1, 0), by elementary computation and Lemma 1, we have Similarly,   (  (, , 0, 1)) = 2 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   () = 2 2 − 6 + 4 + 8.
With the analogous method with Theorem 3, we can deduce the minimum Wiener polarity index of 4-uniform (see the proof of the Appendix).
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  ≥ 5.For the case of extremal cycle-block with  ≥ 3 of Wiener polarity index is very complex, we do not discuss it here.

Figure 1 :
Figure 1: A chain hexagonal cactus (a) and a star cactus graph (b).