Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders

Abstract Let G be a connected graph and u and v two vertices of G. The hyper-Wiener index of graph G is WW(G)=12∑u,v∈V(G)(dG(u,v)+dG2(u,v)) $\begin{array}{} WW(G)=\frac{1}{2}\sum\limits_{u,v\in V(G)}(d_{G}(u,v)+d^{2}_{G}(u,v)) \end{array}$, where dG(u, v) is the distance between u and v. In this paper, we first give the recurrence formulae for computing the hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. We then obtain the sharp upper and lower bounds for the hyper-Wiener index among polyphenyl chains and polyphenyl spiders, respectively. Moreover, the corresponding extremal graphs are determined.


Introduction
Let G be a graph with vertex set V(G) and edge set E(G). The distance d G (u, v) between vertices u and v is the number of edges on a shortest path connecting these vertices in G. Let u ∈ V(G). Denoted by D G (u) is the sum of the distances between u and all other vertices of G.
The Wiener index [1] of G is de ned as the sum of distances between all pairs of vertices in G, i.e., The hyper-Wiener index of G, denoted by WW(G), is de ned as where the summation goes over all pairs of vertices in G. For two vertices u and v of G, set α G (u, v) = d G (u, v)(d G (u, v) + ) and A G (u) = v α G (u, v), where this summation extends to all the vertices di erent from u. Then (1) is expressed as follows.
The hyper-Wiener index, which was rst proposed by Milan Randić [2], is introduced as one of the distancebased molecular structure descriptors. Klein et al. [3] extended Randić's de nition as a generalization of the Wiener index for all connected graphs. For more studies on hyper-Wiener index, see , among others.
The polyphenyl system with n hexagons is obtained from two adjacent hexagons that are sticked by a path. Polyphenyl systems are of great importance for theoretical chemistry because they are natural molecular graph representations of benzenoid hydrocarbons [26].
A polyphenyl system is called a polyphenyl chain PCn with n hexagons [4,26], and it can be regarded as a polyphenyl chain PC n− with n − hexagons adjoining to a new terminal hexagon by a cut edge, the resulting graph see Figure 1. Let PCn = B B · · · Bn be a polyphenyl chain with n(n ≥ ) hexagons, where B i is the i-th hexagon of PCn attached to B i− by a cut edge u i− c i , i = , , · · · , n. A vertex v of H i is said to be ortho-, meta-and paravertex of H i if the distance between v and c i is 1, 2 and 3, denoted by o i , m i and p i , respectively. In particular, A polyphenyl spider, denoted by PS(r, s, t), is obtained by three nonadjacent vertices of a hexagon B joining a polyphenyl chain PC i (i = r, s, t), respectively, the resulting graph see Figure 2. In particular, the hexagon B is called the center of PS(r, s, t), and three components of PS(r, s, t) deleting the center B are called legs of PS(r, s, t). A polyphenyl spider is called a polyphenyl ortho-spiedr if every leg of the polyphenyl spider is a polyphenyl ortho-chain. A polyphenyl spider is called a polyphenyl meta-spiedr if every leg of the polyphenyl spider is a polyphenyl meta-chain. A polyphenyl spider is called a polyphenyl para-spiedr if every leg of the polyphenyl spider is a polyphenyl para-chain. Clearly, a polyphenyl spider is a polyphenyl system. In this paper, we mainly investigate the properties of hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. The rest of this paper is organized as follows. In Section 2, we present some properties of hyper-Wiener index of polyphenyl chains, and give the lower and upper bounds on the hyper-Wiener index among polyphenyl chains. In Section 3, we will give some properties of hyper-Wiener index of polyphenyl spiders, and the extremal polyphenyl spiders with respect to the hyper-Wiener index are obtained.

Hyper-Wiener index of polyphenyl chains
In this section, we will investigate some properties of hyper-Wiener index of polyphenyl chains. Theorem 2.1. Let PCn be a polyphenyl chain with n(n ≥ ) hexagons and u n− cn a cut edge of PCn(see Figure  1). Then Proof. By Eq. (2), we obtain that Simplifying M, we have By (1) and de nitions of A G (u) and D G (u), we have WW(C ) = , A C (cn) = and D C (cn) = . By (9) and (10), we obtain that The proof is completed.

Lemma 2.2. Let PSPn(n ≥ ) be a polyphenyl para-chain with n hexagons. Then
Proof. By the de nition of D G (u), we have Similarly, by the de nition of A G (u), we obtain that By Theorem 2.1, (5) and (6)

Lemma 2.3. Let PSOn(n ≥ ) be a polyphenyl ortho-chain with n hexagons. Then
Proof. By the de nition of D G (u), we have = n − n + .
Similarly, by the de nition of A G (u), we have
The proof is completed.
We shall use T (r, s, t) to denote the set of all polyphenyl spiders with three legs of lengths r, s, t.  PS(r, s, t) is generated by PS(r − , s, t).
Checking PS(r − , s, t), we know that d PS(r− ,s,t) (u, x) ≤ d PS(r− ,s,t) (u, y) ≤ d PS(r− ,s,t) (u, z), where u is any vertex of PS(r − , s, t), and x, y, z is a ortho-, meta-and para-vertex of B r− in leg PC(r − ). This implies, by the de nitions A G (u) and D G (u), that A PS(r− ,s,t) (x) < A PS(r− ,s,t) (y) < A PS(r− ,s,t) (z) and D PS(r− ,s,t) (x) < D PS(r− ,s,t) (y) < D PS(r− ,s,t) (z). By the de nition of a polyphenyl spider, PS(r, s, t) can be obtained from PS(r − , s, t) by attaching a hexagon Br through three attaching. We use PS o (r, s, t) to denote PS(r, s, t) obtained from PS(r − , s, t) by attaching a hexagon Br to ortho-vertex of B i− in PC r− . And PS m (r, s, t) denotes PS(r, s, t) obtained from PS(r − , s, t) by attaching a hexagon Br to meta-vertex of B i− in PC r− . And PS p (r, s, t) denotes PS(r, s, t) obtained from PS(r− , s, t) by attaching a hexagon Br to para-vertex of B i− in PC r− . By Theorem 3.1, we obtain that WW (PS o (r, s, t)) < WW(PS m (r, s, t)) < WW (PS p (r, s, t)). By the de nition of PS(r, s, t), the theorem holds.
Next we shall introduce a graph operation that can be considered as graph transformations, and we shall show that generally, the transformed graph will have larger permanental sum than that of the original graph.
The proof is completed.
By repeated applications of Transformation I, we can obtain the following result. De nition 3.6. Let PSP(r, s, t) be a polyphenyl para-spider and r ≤ s ≤ t. The polyphenyl para-spider PSP(r − , s, t+ ) is obtained from PSP(r, s, t) by deleting the last hexagon Br of the leg PCr in PSP(r, s, t) and attaching Br to para-vertex of B t in leg PC t . We de ne the transformation from PSP(r, s, t) to PSP(r − , s, t + ) as type II. Lemma 3.7. Let PSP(r, s, t) and PSP(r − , s, t + ) be two polyphenyl para-spiders and r ≤ s ≤ t. Then WW(PSP(r, s, t)) < WW (PSP(r − , s, t + )). For any vertex x of leg PCPs in PSP(r − , s, t), since r − < r ≤ t, d PSP(r− ,s,t) (u r− , x) < d PSP(r− ,s,t) (u t , x). By the de nitions of A G (u) and D G (u), we obtain that A PSP(r− ,s,t) (u t ) > A PSP(r− ,s,t) (u r− ) and D PSP(r− ,s,t) (u t ) > D PSP(r− ,s,t) (u r− ). Thus WW (PSP(r, s, t)) − WW(PSP(r − , s, t + )) > .
By repeated applications of Transformation II, we can obtain a result as follows. Proof. By Theorem 3.2 and Lemmas 3.5 and 3.8, the proof of Theorem 3.9 is straightforward.