On the peripheral Wiener index of graphs

Discrete Applied Mathematics 258 (2019) 135–142

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On the peripheral Wiener index of graphs Hongbo Hua Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai’an, Jiangsu 223003, PR China

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a b s t r a c t The (ordinary) Wiener index of a connected graph is defined to be the sum of distances between all vertex pairs in this graph. For a connected graph, its peripheral Wiener index is defined as the sum of distances between all pairs of peripheral vertices (vertices whose eccentricities are equal to diameter). More recently, Chen et al. (2018) investigated extremal problems on the peripheral Wiener index for general trees and trees with given restricted conditions, respectively. In this paper, we obtain further results on the peripheral Wiener index. First, we give sharp lower and upper bounds on the peripheral Wiener index for graphs without cut vertices. Second, we give two sharp upper bounds on the peripheral Wiener index in terms of other distance-based graph invariants. Finally, we establish sharp lower and upper bounds on the difference between the Wiener index and peripheral Wiener index for general connected graphs. © 2018 Elsevier B.V. All rights reserved.

Article history: Received 23 June 2018 Received in revised form 19 November 2018 Accepted 26 November 2018 Available online 18 December 2018 Keywords: Wiener index Peripheral Wiener index Diameter 2-connected graph Bounds

1. Introduction Throughout this paper we consider only simple connected graphs. For a graph G = (V , E) with vertex set V = V (G) and edge set E = E(G), let dG (u, v ) be the distance between vertices u and v in G. The eccentricity of a vertex v in a graph G is defined to be εG (v ) = max{dG (u, v )|u ∈ V (G)}. The diameter of a connected graph G is equal to max{εG (v )|v ∈ V (G)}. A connected graph is said to be a self-centered graph if each vertex is a center in this graph, that is, the eccentricity of each vertex is equal to the diameter. For a graph G with diameter d, let v be a vertex in G, if εG (v ) = d, then v is said to be a peripheral vertex of G. Let P(G) be the set of peripheral vertices in G. A path in a connected graph is said to be a diametral path, if this path is of length equal to the diameter. A connected graph is said to be a tree if it contains no cycles. Let Pn , Cn and Kn be the path, cycle and complete graph of order n, respectively. Let v be a vertex in a connected graph G. If G − v has at least two components, then G is said to be a 1-connected graph and v is called a cut vertex of G. If G contains no cut vertices, then G is said to be a 2-connected graph. In particular, the graph P2 has no cut vertices, but it is not 2-connected. For other notation and terminology not defined here, the readers are referred to [2]. A topological descriptor is a numerical descriptor of the topology of a molecule. These topological descriptors are used for prediction of the physico-chemical and/or biological properties of molecules in quantitative structure–property relationship (QSPR) and quantitative structure–activity relationship (QSAR) studies [19]. One of the oldest distance-based topological descriptors or graph invariants associated with a connected graph G is the Wiener index, which is defined [20] as the sum of distances over all unordered vertex pairs in G, namely, W (G) =

∑

dG (u, v ) =

{u, v}⊆V (G)

1 ∑ 2

v∈V (G)

E-mail address: [email protected]. https://doi.org/10.1016/j.dam.2018.11.031 0166-218X/© 2018 Elsevier B.V. All rights reserved.

DG (v )

136

H. Hua / Discrete Applied Mathematics 258 (2019) 135–142

where DG (v ) = u∈V (G) dG (u, v ). For recent results on Wiener index of graphs, see [4–7,9–14,16] and the references cited therein. More recently, Chen et al. [3] investigated a new distance-based graph invariant, named peripheral Wiener index in [17], which is defined for a connected graph G as

∑

∑

PW (G) =

dG (u, v ).

{u, v}⊆P(G)

Clearly, we can rewrite the peripheral Wiener index as PW (G) =

1 ∑ 2

DP(G) (u),

(1)

u∈P(G)

where DP(G) (u) = v∈P(G) dG (u, v ). In [3], Chen et al. considered the extremal problems for the peripheral Wiener index of trees. More specifically, they first determined the trees with the minimum and the maximum peripheral Wiener index in general trees, respectively. Then they gave sharp lower and upper bounds on the peripheral Wiener index among all trees with given number of peripheral vertices. Finally, they determined the trees with the minimum and the maximum peripheral Wiener index among all trees with given diameter, respectively. In this paper, we obtain further results of the peripheral Wiener index. We organize this paper as follows. In Section 2, we give sharp lower and upper bounds on the peripheral Wiener index for 2-connected graphs. In Section 3, we give two sharp upper bounds on the peripheral Wiener index in terms of other distance-based graph invariants. In Section 4, we establish sharp lower and upper bounds on the difference between Wiener index and peripheral Wiener index for general connected graphs. In the Section 5, we summarize our results concerning the peripheral Wiener index.

∑

2. Sharp bounds on the peripheral Wiener index for 2-connected graphs In this section, we consider bounds on the peripheral Wiener index of the 2-connected graphs. Note that a 2-connected graph has at least three vertices, and the unique 2-connected graph with three vertices is just the cycle C3 . So, we may assume that all 2-connected graphs under consideration have at least four vertices. We need a result due to Plesník [18] reported in 1984. Theorem 2.1. Let G be a 2-connected graph of order n. Then

W (G) ≤

⎧ ⎪ ⎪ ⎨

n3

8 ⎪ ⎪ n(n − 1)(n + 1)

⎩

8 with equality if and only if G ∼ = Cn .

if n is even, if n is odd,

Let Kn − e be the graph obtained from Kn by deleting any one edge e. We first establish sharp lower and upper bounds on the peripheral Wiener index of 2-connected graphs. Theorem 2.2. Let G be a 2-connected graph of order n ≥ 4. Then

2 ≤ PW (G) ≤

⎧ ⎪ ⎪ ⎨

n3

8 ⎪ ⎪ n(n − 1)(n + 1)

if n is even, (2)

if n is odd, 8 where the left-hand side equality holds if and only if G ∼ = Kn − e, while the right-hand side equality holds if and only if G ∼ = Cn .

⎩

Proof. Let d be the diameter of G. We first prove the lower bound. According to the definition of the peripheral Wiener index, we have W (G) ≥ PW (G) with equality if and only if P(G) = V (G). Since |P(G)| ≥ 2 for any connected graph G, we have n(n−1) PW (G) ≥ d. If d ≥ 3, then PW (G) ≥ d ≥ 3 > 2, as claimed. If d = 1, then G ∼ = Kn , and thus, PW (G) = W (G) = 2 > 2 for n ≥ 4. If d = 2 and |P(G)| ≥ 3, then PW (G) ≥ 3 > 2. So, PW (G) = 2 only if d = 2 and |P(G)| = 2. In this case, let P(G) = {u, v}. Then εG (u) = εG (v ) = 2, and εG (x) = 1 for each x ∈ V (G) \ {u, v}. So, G ∼ = Kn − e. Conversely, if G ∼ = Kn − e, then we have PW (G) = 2. Now, we prove the upper bound. By Theorem 2.1, we have PW (G) ≤ W (G)

≤

⎧ ⎪ ⎪ ⎨

(3)

n3

8 ⎪ ⎪ n(n − 1)(n + 1)

⎩

8

if n is even, (4) if n is odd.

H. Hua / Discrete Applied Mathematics 258 (2019) 135–142

137

Thus, the right-hand side inequality of (2) follows. Now, we consider the equality case. If the right-hand side equality of (2) is attained, then both (3) and (4) become equalities. The equality in (3) is attained if and only if P(G) = V (G), i.e., G is a self-centered graph. The equality in (4) is attained if and only if G ∼ = Cn . Summarizing above, the right-hand side equality in (2) is attained if and only if G ∼ = Cn . □ 3. Sharp bounds on the peripheral Wiener index in terms of other graph invariants In this section, we consider bounds on the peripheral Wiener index in terms of other graph invariants. We give two sharp upper bounds on the peripheral Wiener index in terms of other distance-based graph invariants. Theorem 3.1. Let G be a connected graph of order n with k peripheral vertices. If v is a vertex chosen in P(G) such that DP(G) (v ) = minx∈P(G) {DP(G) (x)}, then PW (G) ≤ (k − 1)DP(G) (v ) with equality if and only if k = 2. Proof. When n = 2, it is easy to check that the theorem holds. So, we may assume that n ≥ 3. Clearly, k ≥ 2. By our choice of v , for any given vertex u ∈ P(G) \ {v} and any other vertex w ∈ P(G) \ {u, v},1 it can be seen that dG (u, w ) − dG (v, w ) ≤ dG (u, v ).

(5)

Now,

∑

∑

DP(G) (u) =

u∈P(G), u̸ =v

u∈P(G), u̸ =v

DP(G) (v )

u∈P(G), u̸ =v

[

∑

=

∑

(DP(G) (u) − DP(G) (v )) +

∑

∑

dG (u, w ) −

u∈P(G), u̸ =v w∈P(G)\{u, v}

]

dG (v, w ) +

w∈P(G)\{u, v}

(k − 1)DP(G) (v )

∑

∑

u∈P(G), u̸ =v

w∈P(G)\{u, v}

≤ = (k − 2)

∑

dG (u, v ) + (k − 1)DP(G) (v )

(by (5))

dG (u, v ) + (k − 1)DP(G) (v )

u∈P(G), u̸ =v

= (2k − 3)DP(G) (v )

∑

(as

dG (u, v ) = DP(G) (v )).

u∈P(G), u̸ =v

From the above, we have

∑

DP(G) (u) ≤ (2k − 2)DP(G) (v ).

u∈P(G)

Thus PW (G) ≤ (k − 1)DP(G) (v ).

(6)

Now suppose that the equality holds in (6). Then all the inequalities in the proof of inequality (6) must be equalities. So, we must have that for all u ∈ P(G) \ {v} and all w ∈ P(G) \ {u, v}, the equality in (5) is attained. In this case, we claim that k = 2. Suppose to the contrary that k ≥ 3. Let u be a vertex in G such that dG (u, v ) = d, where d is the diameter of G. Then u ∈ P(G). So, for any w ∈ P(G) \ {u, v}, we have dG (u, w ) − dG (v, w ) ≤ d − 1 < d = dG (u, v ), a contradiction to our assumption that the equality in (5) is attained for each u in V (G) \ {v} and all w ∈ P(G) \ {u, v}. So, k = 2. Conversely, if k = 2, then PW (G) = d = (k − 1)DP(G) (v ). This completes the proof. □ The Harary index (see e.g. [11]) of a connected graph G, denoted by H(G), is defined to be H(G) =

∑ {u, v}⊆V (G), u̸=v

1 dG (u, v )

.

In the following, we give a sharp upper bound on the peripheral Wiener index in terms of Harary index. 1 If k = 2, then P(G) = {u, v}, i.e. P(G) \ {u, v} = ∅. In this case, we assume that d (u, w ) = 0 = d (v, w ) on the left-hand side of the inequality (5). G G So, (5) holds for the case when k = 2.

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H. Hua / Discrete Applied Mathematics 258 (2019) 135–142

Theorem 3.2. Let G be a connected graph of order n and size m with diameter d. Then PW (G) ≤ d2 H(G) − m(d2 − 1) with equality if and only if G ∼ = Kn or C4 . Proof. Let γ (G; k) be the number of vertex pairs at distance k in G. Recall that H(G) = ∑ k≥1 kγ (G; k). Then

( 2

2

d · H(G) = d

m+

∑ γ (G; k) 2≤k≤d

≥ d2 m +

∑

∑

k≥1

γ (G;k) k

and that W (G) =

)

k

kγ (G; k)

(7)

2≤k≤d

= W (G) + m(d2 − 1). It is obvious that the equality holds in (7) if and only if d ≤ 2. Since PW (G) ≤ W (G) with equality if and only if G is a self-centered graph, we have PW (G) ≤ W (G) ≤ d2 H(G) − m(d2 − 1) with equality if and only if G is a self-centered graph and d ≤ 2, i.e., G ∼ = Kn or C4 . This completes the proof. □ 4. The difference between the Wiener index and peripheral Wiener index In this section, we investigate the difference between the Wiener index and peripheral Wiener index for general connected graphs. We will establish sharp lower and upper bounds on the difference between the Wiener index and peripheral Wiener index for general connected graphs. We first introduce some preliminary results. Lemma 4.1. Suppose that G is a connected graph of order n and diameter d. If u is a vertex in G with εG (u) = d, then G − u is connected. Proof. Suppose to the contrary that G − u = G1 ∪ · · · ∪ Gs (s ≥ 2) and |Gi | ≥ 1 for each 1 ≤ i ≤ s. Let ⋃v be a vertex chosen in G such that dG (u, v ) = d. Also, we may suppose that v ∈ V (G1 ). Then for any vertex w ∈ si=2 V (Gi ), we have d ≥ dG (v, w ) = dG (v, u) + dG (u, w ) ≥ dG (v, u) + 1 > d, a contradiction. This completes the proof. □ Before we proceed any further, we introduce a well-known result on connectivity of a graph due to Menger [15] in 1927. Theorem 4.2 (Menger [15]). Let G be a graph and u, v be two distinct nonadjacent vertices of G. Then the maximum number of pairwise internally vertex disjoint paths connecting u and v is equal to the minimum number of vertices in a vertex cut set that separates u and v . Definition 4.1. Suppose that G is a connected graph with diameter d and Pd+1 = v0 v1 . . . vd is a diametral path in G. A vertex vi (1 ≤ i ≤ d − 1) is said to be a separating vertex of the diametral path Pd+1 , if v0 and vd belong to different components of G − {vi }. Remark 4.2. According to Definition 4.1, when we refer to a separating vertex, it is always associated with a diametral path. A separating vertex of a diametral path in a graph is necessarily a cut vertex of this graph. Conversely, it is not necessarily true. To illustrate this, we see an example in Fig. 1. In what follows, when we mention a diametral path Puv , we mean that this diametral path whose both end-vertices are u and v , respectively. From Fig. 1, we know that both u and v are separating vertices of the diametral path Pxy = xuv1 wv2 v y. The vertex w is not a separating vertex of the diametral path Pxy = xuv1 wv2 v y, but w is a cut vertex of G. Lemma 4.3. Suppose that G is a 1-connected graph of order n and diameter d. If d ≥ containing a separating vertex.

n , 2

then there exists a diametral path

H. Hua / Discrete Applied Mathematics 258 (2019) 135–142

139

Fig. 1. A graph used to illustrate the separating vertex.

Proof. Since G is a 1-connected graph of order n, we must have n ≥ 3. Suppose that w is a cut vertex in G. Let G − w = G1 ∪ · · · ∪ Gs (s ≥ 2) and |Gi | ≥ 1 for each 1 ≤ i ≤ s. Suppose to the contrary that each diametral path in G does not contain a separating vertex. Now, if Puv is a diametral path such that u ∈ V (Gi ) and v ∈ V (Gj ), then i = j, for otherwise, w is a separating vertex of Puv , a contradiction. Moreover, Puv lies entirely within some Gk (1 ≤ k ≤ s). Since Puv does not contain a separating vertex, by Theorem 4.2, there are at least 1 two internally vertex disjoint paths connecting u and v in Gk . As s ≥ 2, we have n ≥ 2d + 1, that is, d ≤ n− , a contradiction. 2 This completes the proof. □ Lemma 4.4. Suppose that G is a 1-connected graph. If Puv is a diametral path which contains a separating vertex, then G − u − v is connected. Proof. Let d be the diameter of G. Since G is 1-connected, we have d ≥ 2. We shall proceed by contradiction. Suppose that G − u − v is disconnected. By Lemma 4.1, both G − u and G − v are connected. According to our assumption that G − u − v is disconnected, v must be a cut vertex in G − u. Let (G − u) −v = H1 ∪· · ·∪ Hl (l ≥ 2) and |Hi | ≥ 1 for each 1 ≤ i ≤ l. Since Puv is a diametral path and d ≥ 2, uv ̸ ∈ E(G). So, NG (u) ∩ V (Hi ) ̸ = ∅ for each 1 ≤ i ≤ l, for otherwise, we can deduce that v is also a cut vertex in G, a contradiction. Moreover, V (Hi ) ∩ V (Hj ) = ∅ for any 1 ≤ i, j ≤ l and i ̸ = j. So, there exist at least two internally vertex disjoint paths connecting u and v in G, a contradiction to our assumption that Puv is a diametral path which contains a separating vertex. This completes the proof. □ We recall a well-known result for the Wiener index of trees. Theorem 4.5 ([8]). Let T be a tree of order n. Then

( W (T ) ≤

)

n+1 3

with equality if and only if T ∼ = Pn . For a connected graph G, the remoteness of G, denoted by ρ (G), is defined as

ρ = ρ (G) = max

v∈V (G)

1 n−1

DG (v ).

We need a result on the remoteness due to Aouchiche and Hansen, which reads as follows. Theorem 4.6 ([1]). Let G be a connected graph of order n with remoteness ρ . Then

ρ≤

n 2

with equality if and only if G ∼ = Pn . Now, we state and prove our main result of this section. Theorem 4.7. Let G be a connected graph of order n. Then 0 ≤ W (G) − PW (G) ≤

1 6

(n − 2)(n − 1)(n + 3),

where the left-hand side equality holds if and only if G is a self-centered graph, while the right-hand side equality holds if and only if G ∼ = Pn . Proof. When n = 2, the result is obvious. So we assume that n ≥ 3. Let d be the diameter of G. The left-hand side inequality is obvious, as P(G) ⊆ V (G). Moreover, the left equality is attained if and only if P(G) = V (G), that is, G is a self-centered graph. Now, we consider the right-hand side inequality. We should complete the proof by considering the following three cases.

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H. Hua / Discrete Applied Mathematics 258 (2019) 135–142

Case 1. G is a 2-connected graph. In this case, we have G ∼ = C3 , or C4 , or K4 , then V (G) = P(G), and = C3 if n = 3, and G ∼ = C4 , or K4 , or K4 − e if n = 4. If G ∼ thus, W (G) − PW (G) = 0 <

(n − 2)(n − 1)(n + 3) 6

.

If G ∼ = K4 − e, then (n − 2)(n − 1)(n + 3)

W (G) − PW (G) = 5 < 7 =

6

.

Now, we may assume that n ≥ 5. By Theorem 2.1 and the fact that PW (G) > 0, we have

W (G) − PW (G) <

⎧ ⎪ ⎪ ⎨

n3

⎩

if n is odd.

8

It is easy to check that for n ≥ 5, we have W (G) − PW (G) <

if n is even,

8 ⎪ ⎪ n(n − 1)(n + 1) n3 8

<

(n − 2)(n − 1)(n + 3) 6

(n−2)(n−1)(n+3) 6

and

n(n−1)(n+1) 8

<

(n−2)(n−1)(n+3) . 6

Thus, if n ≥ 5, then

.

So, the upper bound follows readily upon this case. Case 2. G is a 1-connected graph of order n and diameter d with d ≥

n . 2

In this case, by Lemmas 4.1, 4.3 and 4.4, G has a diametral path, say Puv , such that all three subgraphs G − u, G −v , G − u −v are connected. Now, we have

∑

W (G) − PW (G) = dG (u, v ) +

{x, y}⊆V (G)\{u, v}

∑

∑

dG (x, y) +

dG (u, x) +

x∈V (G)\{v}

dG (v, x) − PW (G)

x∈V (G)\{u}

∑

≤ d+

{x, y}⊆V (G)\{u, v}

∑

∑

dG (x, y) +

dG (v, x) − d

dG (u, x) +

x∈V (G)\{v}

(as PW (G) ≥ d)

(8)

x∈V (G)\{u}

=

∑

dG (x, y) +

{x, y}⊆V (G)\{u, v}

∑

∑

dG (u, x) +

x∈V (G)\{v}

dG (v, x).

(9)

x∈V (G)\{u}

Since G − u, G − v and G − u − v are all connected, we have (a) dG (x, y) ≤ dG−u−v (x, y)

(10)

for each vertex pair {x, y} ⊆ V (G) \ {u, v}; (b) dG (u, x) ≤ dG−v (u, x)

(11)

for each vertex x ∈ V (G) \ {v}; and (c) dG (v, x) ≤ dG−u (v, x) for each vertex x ∈ V (G) \ {u}.

(12)

H. Hua / Discrete Applied Mathematics 258 (2019) 135–142

141

Thus, by (10) and Theorem 4.5,

∑

∑

dG (x, y) ≤

{x, y}⊆V (G)\{u, v}

dG−u−v (x, y)

(13)

{x, y}⊆V (G)\{u, v}

= W (G − u − v ) ( ) (n − 2) + 1 ≤ =

3 (n − 1)(n − 2)(n − 3)

(14)

.

(15)

6 By the definition of the remoteness, (11) and Theorem 4.6,

∑

dG (u, x) ≤

x∈V (G)\{v}

∑

dG−v (u, x)

(16)

x∈V (G)\{v}

= DG−v (u) ≤ (n − 2)ρ (G − v ) n−1 ≤ (n − 2) · .

(17) (18)

2 By the definition of the remoteness, (12) and Theorem 4.6,

∑

dG (v, x) ≤

x∈V (G)\{u}

∑

dG−u (v, x)

(19)

x∈V (G)\{u}

= DG−u (v ) ≤ (n − 2)ρ (G − u) n−1 ≤ (n − 2) · .

(20) (21)

2

By (9), (15), (18), (21), we have W (G) − PW (G) ≤

=

(n − 1)(n − 2)(n − 3)

+ (n − 1)(n − 2)

6 (n − 1)(n − 2)(n + 3)

(22)

.

6 Now, we consider the equality case. (n−1)(n−2)(n+3) , then (22) becomes equality. So, all the inequalities (8), (13), (14), (16)–(21) become If W (G) − PW (G) = 6 equalities. The equality in (8) is attained only if PW (G) = d. The equality in (13) is attained only if for each vertex pair {x, y} ⊆ V (G) \ {u, v}, the shortest path connecting x and y in G does not pass through u and v . The equality in (16) is attained only if for each vertex x ∈ V (G) \ {v}, the shortest path connecting x and u in G does not pass through v . The equality in (19) is attained only if for each vertex x ∈ V (G) \ {u}, the shortest path connecting x and v in G does not pass through u. The equality in (14) is attained only if G − u − v ∼ = Pn−2 . D (u) The equality in (17) is attained only if Gn−v = ρ (G − v ). −2 By Theorem 4.6, the equality in (18) is attained only if G − v ∼ = Pn−1 , that is, dG (u) = 1 (as uv ̸∈ E(G)). D (v ) The equality in (20) is attained only if Gn−−u2 = ρ (G − u). By Theorem 4.6, the equality in (21) is attained only if G − u ∼ = Pn−1 , that is, dG (v ) = 1. Summarizing above, we have G ∼ = Pn . Conversely, if G ∼ = Pn , then W (Pn ) − PW (Pn ) =

(n − 1)(n − 2)(n + 3)

6 Thus, the upper bound holds for this case.

.

Case 3. G is a 1-connected graph of order n and diameter d with d <

n . 2

We further consider the following two subcases. Subcase 3.1. G has a diametral path, say Puv , such that Puv contains a separating vertex. By Lemma 4.4, G − u − v is connected. Moreover, by Lemmas 4.1 and 4.3, both subgraphs G − u and G − v are also connected. Similar to Case 2, we can prove that W (G) − PW (G) <

(n − 1)(n − 2)(n + 3) 6

.

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H. Hua / Discrete Applied Mathematics 258 (2019) 135–142

Subcase 3.2. Any one diametral path in G does not contain a separating vertex. Let Puv be a diametral path in G. By Theorem 4.2, there exist two internally vertex disjoint paths connecting u and v . Let x be a cut vertex in G. Set G − x = G1 ∪ · · · ∪ Gs (s ≥ 2). Since Puv does not contain a separating vertex, then the diametral path Puv must lie entirely within some connected component, say G1 , of G − x. Obviously, V (G2 ) ∪· · ·∪ V (Gs ) contains a vertex, say z, which is not a cut vertex. To see this, we let A = {p is a cut vertex in ⋃s G|p ∈ i=2 V (Gi )}, and q ∈ A such that dG (u, q) = max{dG (u, p)|p ∈ A}. Set G − q = H1 ∪ · · · ∪ Ht (t ≥ 2) and |Hi |≥ 1 for each 1 ≤ i ≤ t. We may assume that u ∈ V (H1 ). Similar to our previous discussion, we have ( ) V( (Puv ) ⊆ V (H1)). By our choice of q, any vertex in V (H2 ) ∪ · · · ∪ V (Ht ) cannot be a cut vertex. Moreover,

⋃

2≤i≤s

V (Gi ) ∩

⋃

2≤i≤l

V (Hi )

̸= ∅. So, such

vertex z exists indeed. We claim that G − u − z is connected. By contradiction. Suppose to the contrary that G − u − z = K1 ∪ · · · ∪ Ks (s ≥ 2). Note that G − u is connected by Lemma 4.1. So, z is a cut vertex of G − u. If NG (u) ∩ V (Ki ) = ∅ for some 1 ≤ i ≤ s, we can deduce that z is also a cut vertex in G, a contradiction. So, NG (u) ∩ V (Ki ) ̸ = ∅ for each 1 ≤ i ≤ s. Moreover, V (Ki ) ∩ V (Kj ) = ∅ for any 1 ≤ i, j ≤ s and i ̸ = j. Thus, there exist at least two internally vertex disjoint paths connecting u and z in G. Recall that z ∈ V (G2 ) ∪ · · · ∪ V (Gs ) in G − x and u ∈ V (G1 ). This is a contradiction. So, G − u − z is connected. Clearly, dG (u, z) < d, for otherwise, G has a diametral path Puz which contains a separating vertex x, a contradiction. Since G − u, G − z and G − u − z are all connected, we can prove that W (G) − PW (G) <

(n − 1)(n − 2)(n + 3)

6 by a similar way to Case 2. Thus, the upper bound follows in this case. This completes the proof. □ 5. Concluding remarks In this paper, we explored further properties of the peripheral Wiener index. More specifically, we gave sharp lower and upper bounds on the peripheral Wiener index for graphs without cut vertices, and two sharp upper bounds on the peripheral Wiener index in terms of other distance-based graph invariants. In addition, we established sharp lower and upper bounds on the difference between the Wiener index and peripheral Wiener index for general connected graphs. It may be interesting to investigate the relationships between the peripheral Wiener index and other graph invariants. Also, it remains an open problem to characterize the graph with the maximum peripheral Wiener index among all connected graphs. Acknowledgments This research was supported by National Natural Science Foundation of China under Grant No. 11571135 and Qing Lan Project of Jiangsu Province, P.R. China. References [1] M. Aouchiche, P. Hansen, Proximity and remoteness in graphs: results and conjectures, Networks 58 (2011) 95–102. [2] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan London and Elsevier, New York, 1976. [3] Y.-H. Chen, H. Wang, X.-D. Zhang, Peripheral Wiener index of trees and related questions, Discrete Appl. Math. available online at: https://doi.org/10. 1016/j.dam.2018.05.024. [4] K.C. Das, I. Gutman, On Wiener and multiplicative Wiener indices of graphs, Discrete Appl. Math. 206 (2016) 9–14. [5] A.A. Dobrynin, Hexagonal chains with segments of equal lengths having distinct sizes and the same Wiener index, MATCH Commun. Math. Comput. Chem. 78 (2017) 121–132. [6] A.A. Dobrynin, The Szeged and Wiener indices of line graphs, MATCH Commun. Math. Comput. Chem. 79 (2018) 743–756. [7] A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001) 211–249. [8] R.C. Entringer, D.E. Jackson, D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976) 283–296. [9] L. Feng, X. Zhu, W. Liu, Wiener index, Harary index and graph properties, Discrete Appl. Math. 223 (2017) 72–83. [10] H. Guo, B. Zhou, H. Lin, The Wiener index of uniform hypergraphs, MATCH Commun. Math. Comput. Chem. 78 (2017) 133–152. [11] H. Hua, B. Ning, Wiener index, Harary index and Hamiltonicity of graphs, MATCH Commun. Math. Comput. Chem. 78 (2017) 153–162. [12] S. Klavžar, M.J. Nadjafi-Arani, Improved bounds on the difference between the Szeged index and the Wiener index of graphs, European J. Combin. 39 (2014) 148–156. [13] S. Klavžar, M.J. Nadjafi-Arani, Wiener index in weighted graphs via unification of Θ ∗ -classes, European J. Combin. 36 (2014) 71–76. [14] H. Lei, T. Li, Y. Shi, H. Wang, Wiener polarity index and its generalization in trees, MATCH Commun. Math. Comput. Chem. 78 (2017) 199–212. [15] K. Menger, Zur allgemeinen Kurventheorie, Fund. Math. 10 (1927) 96–115. [16] S. Mukwembi, T. Vetrik, Wiener index of trees of given order and diameter at most 6, Bull. Aust. Math. Soc. 89 (2014) 379–396. [17] K. Narayankar, L.S.B, Peripheral Wiener index of a graph, preprint. [18] J. Plesník, On the sum of all distances in graph or diagraph, J. Graph Theory 8 (1984) 1–24. [19] N. Trinajstić, Chemical Graph Theory, I/II, CRC Press, Boca Raton, 1983. [20] H. Wiener, Structural determination of paraffin boiling point, J. Amer. Chem. Soc. 69 (1947) 17–20.