The Wiener index of a graph, which is the sum of the distances between all
pairs of vertices, has been well studied. Recently, Sills and Wang in 2012
proposed two conjectures on the maximal Wiener index of trees with a given
degree sequence. This note proves one of the two conjectures and disproves the
other.Comment: 7 page

Jin, Ya-Lei

Zhang, Xiao-Dong

English

arXiv

The Wiener index of a graph, which is the sum of the distances between all pairs of vertices, has been well studied. Recently, Sills and Wang in 2012 proposed two conjectures on the maximal Wiener index of trees with a given degree sequence. This note proves one of the two conjectures and disproves the other

Ya-Lei Jin

Xiao-Dong Zhang

CiteSeerX

On the Two Conjectures of the Wiener Index

On the Two Conjectures of the
Wiener Index ∗
Ya–Lei Jin, Xiao–Dong Zhang†
Department of Mathematics, and Ministry of Education
Key Laboratory of Scientific and Engineering Computing,
Shanghai Jiao Tong University
800 Dongchuan road, Shanghai, 200240, P.R. China
(Received February 19, 2013)
Abstract
The Wiener index of a graph, which is the sum of the distances between all pairs
of vertices, has been well studied. Recently, Sills and Wang in 2012 proposed two
conjectures on the maximal Wiener index of trees with a given degree sequence.
This note proves one of the two conjectures and disproves the other.
1 Introduction
The Wiener index of a molecular graph is one of the most classic and well-known topolog-
ical indices in the molecular graph, which was introduced by and named by Wiener [14]
in 1947. It has been extensively studied by chemists and mathematicians over the past
years, see for instance [2]. In the past decade years, the extremal trees that maximize
or minimize the Wiener index among trees with prescribed maximum degree, diameter,
matching and independence numbers, etc., have been studied (see [5, 9, 15] etc.).
Since the degrees of a molecular graph corresponds to the valences of the atoms, it is
one of the most interesting aspects to consider all trees with a prescribed degree sequence.
Wang [12] and Zhang at al. [15] independently proved the extremal tree that minimizes
the Wiener index is greedy tree through different approaches. Moreover, the extremal
tree that maximizes the Wiener index in this category in [12] is incorrect by pointed out
in [13] and [16]. Therefore it is still open problem.
∗Supported by National Natural Science Foundation of China (No. 11271256).
†Correspondent author: Xiao-Dong Zhang (Email: xiaodong@sjtu.edu.cn)
MATCH 
Communications in Mathematical 
and in Computer Chemistry 
MATCH Commun. Math. Comput. Chem. 70 (2013) 583-589 
                          
                                          ISSN 0340 - 6253  
 
Problem 1.1. Characterize the extremal trees that maximize the Wiener index with pre-
scribed degree sequence.
Zhang et al. [16] provided some part results with less than 7 internal vertices. Cela et
al. [2] provide an efficient algorithm for finding the extremal trees with prescribed degree
sequence. Recently, Sills and Wang [11] further studied the maximal Wiener index and
disclosed some relations between the candidate trees for the maximal Wiener index and
the symmetric Dyck paths.
Let T = (V,E) be a tree of order n. The Wiener index W (T ) of T is defined as
W (T ) :=
∑
{u,v}⊆V
d(u, v),
where d(u, v) is the number of edges in a shortest path from u to v. A nonincreasing
sequence of nonnegative integers π = (d1, d2, · · · , dn) is called graphic if there exists a
simple graph having π as its vertex degree sequence. In particular, if
∑n
i=1 di = 2(n− 1),
then π is graphic and any graph with degree sequence π is tree and let Tπ denote the set
of all trees with degree sequence π. Moreover, if
d1 ≥ d2 ≥ · · · ≥ dk ≥ 2 > dk+1 = dk+2 = · · · = dn = 1,
then b = (b1, . . . , bk) := (d1− 1, . . . , dk− 1) is called the decremented degree sequence [11].
A caterpillar is a tree in which a single path (called Spine) is incident to (or contains)
every edge. For other terminologies and notions, we follow from [1,11]. Since it has been
proved [16] that a tree with maximum Wiener index in Tπ has to be a caterpillar, it is
interesting and important to study the Wiener index of caterpillars. Let T be a caterpillar
of order n with n − k leaves and the non-leaf vertices v1, . . . , vk. Then the Winer index
of T is presented in [16]
W (T ) = (n− 1)2 + q(x) ,
where q(x)is the quadratic form
q(x) =
1
2
k∑
i=1
k∑
j=1
|i− j|xixj = xTAkx , (1)
Ak = (aij) with aij =
1
2
|i − j|, x = (x1, . . . , xk)T , and xi = deg(vi) − 1 for i = 1, . . . , k.
In order to obtain some useful upper bounds for the Wiener index in Tπ, Sills and Wang
-584-
observed the largest eigenvalue of Ak is about to
λmax ≈
√
3k2 − 2
10
. (2)
Further, they disclosed some interesting combinatorial relations to other objects from this
study and proposed the following conjecture.
Conjecture 1.2. [11] Let Ak = (aij) be the k × k matrix with aij = 12 |i − j|. If
Ck(λ) = det(Ak − λIk) is the characteristic polynomial of Ak, then
Ck(λ) = (−1)kλk
(
1− k
4
k−1∑
j=1
j
j + 1
(
k + j
2j + 1
)
λ−j−1
)
. (3)
On the other hand, Silly and Wang [11] characterized all extremal trees that maximize
in all chemical trees with prescribed degree sequence π = (d1, . . . , dn) with 4 ≥ d1 ≥ · · · ≥
dn = 1. This result can be stated as follows:
Theorem 1.3. [11] Let π = (d1, . . . dk, dk+1, . . . , dn) with 4 ≥ d1 ≥ · · · ≥ dk >
dk+1 = · · · = dn = 1 and let b = (b1, . . . , bk) be the decrmented degree sequence. If
{b1, b2, . . . , bk} = {as, . . . , as︸ ︷︷ ︸
ms
, as−1, . . . , as−1︸ ︷︷ ︸
ms−1
, . . . , a1, . . . , a1︸ ︷︷ ︸
m1
} with as > as−1 > · · · > a1,
then q(x) is maximized by
x = {as, . . . , as︸ ︷︷ ︸
ls
, as−1, . . . , as−1︸ ︷︷ ︸
ls−1
, . . . , a1, . . . , a1︸ ︷︷ ︸
m1
, . . . , , as−1, . . . , as−1︸ ︷︷ ︸
rs−1
, as, . . . , as︸ ︷︷ ︸
rs
},
where |li − ri| ≤ 1 and li + ri = mi for i = 2, . . . , s.
Further, they [11] proposed the following conjecture
Conjecture 1.4. [11] When k is much larger than s, for
{b1, b2, . . . , bk} = {as, . . . , as︸ ︷︷ ︸
ms
, as−1, . . . , as−1︸ ︷︷ ︸
ms−1
, . . . , a1, . . . , a1︸ ︷︷ ︸
m1
}
with as > as−1 > · · · > a1, then q(x) is maximized by
x = {as, . . . , as︸ ︷︷ ︸
ls
, as−1, . . . , as−1︸ ︷︷ ︸
ls−1
, . . . , a1, . . . , a1︸ ︷︷ ︸
m1
, . . . , , as−1, . . . , as−1︸ ︷︷ ︸
rs−1
, as, . . . , as︸ ︷︷ ︸
rs
},
where |li − ri| ≤ 1 and li + ri = mi for i = 2, . . . , s.
This note is motivated by the above two conjectures. The rest of the note is organized
as follows: In next Section, we prove Conjecture 1.2; while in Section 3, we disprove
Conjecture 1.4.
-585-
2 Proof of Conjecture 1.2
Before presenting a proof of Conjecture 1.2, we need some notations. Let G = (V,E) be
a connected graph with V = {v1, . . . , vn}, Graham and Pollak [6] introduced the distance
matrix D(G) = (dij) of G with dij = d(vi, vj) arising from a data communication problem.
Graham and Lovász [7] proved that the coefficients of the characteristic polynomial of the
distance matrix of a tree can be expressed in terms of the number of certain subforests of
the tree and conjectured that the sequence of coefficients was unimodal with peak at the
center. Colllins [3] proved that the coefficients for a path on n vertices are unimodal with
peak at (1− 1/
√
5)n. From the context, it is easy to get the following Lemma from [3]
Lemma 2.1. [3] Let Pn be a path of order n and distance matrix D(Pn) = (dij) with
dij = |i− j|, i.e.,
D(Pn) =
⎛
⎜⎜⎝
0 1 2 . . . n− 1
1 0 1 . . . n− 2
. . . . . . . . . . . . . . .
n− 1 n− 2 n− 3 . . . 0
⎞
⎟⎟⎠ .
let δi be the coefficient of λ
i in the distance matrix polynomial det(D(Pn)− λIn). Then
δn = (−1)n, δn−i = (−1)n−1
2i−2n(i− 1)
i
(
n+ i− 1
2i− 1
)
, for i = 1, . . . , n.
Proof. It follows from [3].
Now we are ready to prove Conjecture 1.2
Theorem 2.2. Let Ak = (aij) be the k × k matrix with aij = 12 |i − j|. If Ck(λ) =
det(Ak − λIk) is the characteristic polynomial of Ak, then
Ck(λ) = (−1)kλk
(
1− k
4
k−1∑
j=1
j
j + 1
(
k + j
2j + 1
)
λ−j−1
)
.
Proof. Clearly, Ak =
1
2
D(Pk). Then by Lemma 2.1
Ck(λ) = det(Ak − λI) = det(
1
2
D(Pk)− λI)
= (
1
2
)k det(D(Pk)− (2λ)Ik)
= (
1
2
)k((−1)k(2λ)k + · · ·+ δn−i(2λ)n−i + · · ·+ δ0)
= (−1)kλk + · · ·+ (−1)
k−1(i− 1)k
4i
(
k + i− 1
2i− 1
)
λk−i + · · ·+ (−1)
k−1(k − 1)
4
= (−1)kλk
(
1− k
4
k−1∑
j=1
j
j + 1
(
k + j
2j + 1
)
λ−j−1
)
.
-586-
Hence Theorem 2.2 holds.
On the largest eigenvalue of Ak, there is the following result.
Theorem 2.3. The largest eigenvalue of Ak = (aij) with aij =
1
2
|i− j| is equal to
λmax =
1
2(cosh θ − 1) ,
where θ is the positive solution of tanh( θ
2
) tanh(kθ
2
) = 1
k
. Moreover,
λmax =
k2
4a2
− 2 + a
2
12a2
+ o(
1
n2
),
where a is the root of a tanh(a) = 1, i.e, a ≈ 1.199679.
Proof. It follows from Ak =
1
2
D(Pk), Theorem 2.1 and Corollary 2.2 in [10]
3 Disproof of Conjecture 1.4
In order to disprove Conjecture 1.4, we first present the following result
Theorem 3.1. Let {b1, b2, . . . , bk} = {as, . . . , as︸ ︷︷ ︸
ms
, as−1, . . . , as−1︸ ︷︷ ︸
ms−1
, . . . , a1, . . . , a1︸ ︷︷ ︸
m1
} with as >
as−1 > · · · > a1. If as >
∑s−1
i=1 miai and ms = 2h+ 1, then q(x) is uniquely maximized by
x = {as, . . . , as︸ ︷︷ ︸
h+1
, a1, . . . , a1︸ ︷︷ ︸
m1
, . . . , as−1, . . . , as−1︸ ︷︷ ︸
rs−1
, as, . . . , as︸ ︷︷ ︸
h
}.
Proof. It is easy to see that the assertion hold for s = 2 or k ≤ 5. Now assume that q(x)
is maximized by x = {x1, . . . , xk} for k > 5 and s ≥ 3. Then by Theorem 2.7 in [16],
there exists a 2 ≤ t ≤ k − 2 such that
t−2∑
i=1
xi ≤
k∑
i=t+1
xi,
t−1∑
i=1
xi >
k∑
i=t+2
xi, (4)
and either x1 ≥ · · · ≥ xt, xt ≤ xt+1 ≤ · · · ≤ xk or x1 ≥ · · · ≥ xt−1, xt−1 ≤ xt+1 ≤ · · · ≤
xk. Hence x can be rewritten as
x = {as, as, ..., as︸ ︷︷ ︸
ls
, as−1, as−1, ..., as−1︸ ︷︷ ︸
ls−1
, ..., a1, a1, ..., a1︸ ︷︷ ︸
m1
, ...., as−1, ..., as−1︸ ︷︷ ︸
rs−1
, as, as, ..., as︸ ︷︷ ︸
rs
},
where ls + rs = ms = 2h+ 1. Clearly t > ls and ls > rs. Further we have the following
claim t = ls + 1 and ls = rs + 1. In fact, suppose that t ≥ ls + 2. Then by the condition
of Theorem 3.1,
t−2∑
i=1
xi ≥ lsas ≥ (rs + 1)as > raas +
s−1∑
i=1
miai ≥
k∑
i=t+1
xi,
-587-
which is contradiction to (4). Hence t = ls + 1. Moreover, by (4), we have
(ls − 1)as =
t−2∑
i=1
xi <
k∑
i=t+1
xi ≤ rsas +
s−1∑
i=1
miai < (rs + 1)as.
So ls − 1 < rs + 1, i.e., ls = rs + 1. Therefore the assertion holds.
Remark When k is much larger than s. Let
{b1, . . . , bk} = k + s2, k + s2, k + s2︸ ︷︷ ︸
3
, s− 1, s− 1︸ ︷︷ ︸
2
, s− 2, s− 2︸ ︷︷ ︸
2
, . . . , 2, 2︸︷︷︸
2
, 1, . . . , 1︸ ︷︷ ︸
k−2s+1
},
with k + s2 > s − 1 > · · · > 1. By Theorem 3.1, q(x) is uniquely maximized by x =
(k+ s2, k+ s2, 1, . . . , 1, 2, 2, 3, 3, . . . , s− 1, s− 1, k+ s2). Hence Conjecture 1.4 is not true
for this case.
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