Filomat 26:4 (2012), 683–688
DOI 10.2298/FIL1204683X
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Extremal trees with fixed degree sequence for atom-bond connectivity
index
Rundan Xing
a
, Bo Zhou
a,∗
a
Department of Mathematics, South China Normal University, Guangzhou 510631, China
Abstract. The atom-bond connectivity (ABC) index of a graph G is the sum of
d(u)+d(v)−2
d(u)d(v)
over all edges uv
of G, where d(u) is the degree of vertex u in G. We characterize the extremal trees with fixed degree sequence
that maximize and minimize the ABC index, respectively. We also provide algorithms to construct such
trees.
1. Introduction
Let G be a simple connected graph with vertex set V(G) and edge set E(G). For any vertex v ∈ V(G),
denote by d
G
(v) or d(v) the degree of v in G.
The atom-bond connectivity (ABC) index of G is defined as [1]
ABC(G) =
uv∈E(G)
d(u) + d(v) − 2
d(u)d(v)
.
The ABC index displays an excellent correlation with the heat of formation of alkanes [1], and from it
a basically topological approach was developed to explain the differences in the energy of linear and
branched alkanes both qualitatively and quantitatively [2]. Various properties of the ABC index have been
established, see [3–8].
The (general) Randi
´
c index of a graph G is defined as [9]
R
α
(G) =
uv∈E(G)
(d(u)d(v))
α
,
where α is a nonzero real number. Delorme et al. [10] described an algorithm that determines a tree of fixed
degree sequence that maximizes the (general) Randi
´
c index for α = 1 (also known as the second Zagreb
index [11]). Then Wang [12] characterized the extremal trees with fixed degree sequence that minimize the
(general) Randi
´
c index for α > 0, and maximize the (general) Randi
´
c index for α < 0.
In this note, we use the techniques from [10, 12] to characterize the extremal trees with fixed degree
sequence to maximize and minimize the ABC index.
2010 Mathematics Subject Classification. Primary 05C35; Secondary 05C07, 05C90
Keywords. Atom-bond connectivity index, tree, degree sequence
Received: 27 June 2011; Accepted: 13 September 2011
Communicated by Dragan Stevanovi
´
c
Research supported by the Guangdong Provincial Natural Science Foundation of China (no. S2011010005539)
* Corresponding author
Email addresses: rundanxing@126.com (Rundan Xing), zhoubo@scnu.edu.cn (Bo Zhou)
R. Xing, B. Zhou / Filomat 26:4 (2012), 683–688 684
2. Preliminaries
For a tree T, the degree sequence of T is the sequence of degrees of the non-pendent vertices arranged
in a non-increasing order.
First we give two lemmas.
Let f (x, y) =
x+y−2
xy
for x, y ≥ 1 with x + y > 2.
Lemma 2.1. ([5]) If y ≥ 2 is fixed, then f (x, y) is decreasing in x.
For s > r ≥ 1, let 1
r,s
(x) = f (x, r) − f (x, s).
Lemma 2.2. The fuction 1
r,s
(x) is increasing in x.
Proof. Obviously, 1
r,s
(x) =
1
x
+
1
r
−
2
rx
−
1
x
+
1
s
−
2
sx
. Then
1
′
r,s
(x) =
1
2
rx
r + x − 2
−
1
x
2
+
2
rx
2
−
1
2
sx
s + x − 2
−
1
x
2
+
2
sx
2
=
√
x
x
2
√
r + x − 2
1
√
r
−
√
r
2
−
√
x
x
2
√
s + x − 2
1
√
s
−
√
s
2
.
=
√
x
2x
2
2 − r
r(r + x − 2)
−
2 − s
s(s + x − 2)
.
Let h(t) =
2−t
√
t(t+x−2)
for t ≥ 1 with t + x > 2. It is easily seen that h
′
(t) = −
xt+2(t+x−2)
2(t(t+x−2))
3
2
< 0, implying that h(t) is
decreasing in t. Recall that r < s. Then 1
′
r,s
(x) =
√
x
2x
2
(h(r) − h(s)) > 0, and thus result follows.
For a tree T and i = 0, 1, . . . , let L
i
= L
i
(T) be the set of vertices in T, the minimum distance from which
to the set of pendent vertices of T is i. Clearly, L
0
is exactly the set of pendent vertices in T.
For a graph G with F ⊆ E(G), denote by G − F the subgraph of G obtained by deleting the edges of F.
Similarly, G + W denotes the graph obtained from G by adding edges in W, where W is an subset of edge
set of the complement of G.
3. Upper bound for the ABC index of trees with fixed degree sequence
In this section, we characterize the extremal trees with maximum ABC index among the trees with fixed
degree sequence, and provide an algorithm to construct such trees.
Lemma 3.1. Let T be a tree with maximum ABC index among the trees with fixed degree sequence. Let P =
v
0
v
1
v
2
. . . v
t
be a path in T, where d(v
0
) = d(v
t
) = 1. For 1 ≤ i ≤
t
2
, we may always assume
(i) if i is odd, then d(v
i
) ≥ d(v
t−i
) ≥ d(v
j
) for i + 1 ≤ j ≤ t −i − 1;
(ii) if i is even, then d(v
i
) ≤ d(v
t−i
) ≤ d(v
j
) for i + 1 ≤ j ≤ t −i − 1.
Proof. We argue by induction on i. Suppose that d(v
1
) < d(v
j
) for some 2 ≤ j ≤ t − 2. Let T
′
= T −
{v
0
v
1
, v
j
v
j+1
} + {v
0
v
j
, v
1
v
j+1
}. Obviously, T
′
has the same degree sequence as T. Note that d(v
0
) = 1. Since
j + 1 ≤ t − 1, we have d(v
j+1
) ≥ 2 > 1. Since d(v
j
) > d(v
1
) ≥ 1, we know by Lemma 2.2 that the function
1
d(v
1
),d(v
j
)
(x) is increasing in x, and then
ABC(T) − ABC(T
′
) = f (d(v
0
), d(v
1
)) + f (d(v
j
), d(v
j+1
)) − f (d(v
0
), d(v
j
)) − f (d(v
1
), d(v
j+1
))
=
f (d(v
0
), d(v
1
)) − f (d(v
0
), d(v
j
))
−
f (d(v
1
), d(v
j+1
)) − f (d(v
j
), d(v
j+1
))
= 1
d(v
1
),d(v
j
)
(d(v
0
)) − 1
d(v
1
),d(v
j
)
(d(v
j+1
))
= 1
d(v
1
),d(v
j
)
(1) − 1
d(v
1
),d(v
j
)
(d(v
j+1
)) < 0,
R. Xing, B. Zhou / Filomat 26:4 (2012), 683–688 685
which is a contradiction. Thus d(v
1
) ≥ d(v
j
) for 2 ≤ j ≤ t−2. Similarly, we have d(v
t−1
) ≥ d(v
j
) for 2 ≤ j ≤ t−2.
Thus we may assume that d(v
1
) ≥ d(v
t−1
) ≥ d(v
j
) for 2 ≤ j ≤ t −2. The result for i = 1 follows.
Suppose that the result is true for i = k ≥ 1. We consider the case i = k + 1. Suppose that k is odd. Then
k + 1 is even, and by the induction hypothesis, we have d(v
k
) ≥ d(v
t−k
) ≥ d(v
j
) for k + 1 ≤ j ≤ t − k − 1.
Suppose that d(v
k+1
) > d(v
j
) for some j with k + 2 ≤ j ≤ t −k −2. Let T
′′
= T −{v
k
v
k+1
, v
j
v
j+1
}+ {v
k
v
j
, v
k+1
v
j+1
}.
Obviously, T
′′
has the same degree sequence as T. Note that the path P in T is changed into the path
Q = v
0
v
1
. . . v
k
v
j
v
j−1
. . . v
k+2
v
k+1
v
j+1
v
j+2
. . . v
t
in T
′′
, and the degree of the (k + 1)-th vertex (v
j
) of Q is less than
the degree of the j-th vertex (v
k+1
) of Q in T
′′
. Since j + 1 ≤ t − k − 1, we have d(v
k
) ≥ d(v
j+1
). Similarly as
above, we have
ABC(T) − ABC(T
′′
) = f (d(v
k
), d(v
k+1
)) + f (d(v
j
), d(v
j+1
)) − f (d(v
k
), d(v
j
)) − f (d(v
k+1
), d(v
j+1
))
=
f (d(v
j
), d(v
j+1
)) − f (d(v
k+1
), d(v
j+1
))
−
f (d(v
k
), d(v
j
)) − f (d(v
k
), d(v
k+1
))
= 1
d(v
j
),d(v
k+1
)
(d(v
j+1
)) − 1
d(v
j
),d(v
k+1
)
(d(v
k
)) ≤ 0.
Thus we may assume that d(v
k+1
) ≤ d(v
j
) for k + 2 ≤ j ≤ t−k −2. Similarly, we may also have d(v
t−k−1
) ≤ d(v
j
)
for k + 2 ≤ j ≤ t − k − 2. If d(v
k+1
) > d(v
t−k−1
), then as above, we have ABC(T) ≤ ABC(T −{v
k
v
k+1
, v
t−k−1
v
t−k
}+
{v
k
v
t−k−1
, v
k+1
v
t−k
}). Thus we may assume that d(v
k+1
) ≤ d(v
t−k−1
) ≤ d(v
j
) for k + 2 ≤ j ≤ t − k − 2. The result
follows for i = k + 1 with odd k. Similarly, the result follows for i = k + 1 with even k.
From Lemma 3.1, the following corollary follows easily.
Corollary 3.2. Let T be a tree with maximum ABC index among the trees with fixed degree sequence. For v
i
∈ L
i
and v
j
∈ L
j
with j > i ≥ 1, if i is odd, then d(v
i
) ≥ d(v
j
), and if i is even, then d(v
i
) ≤ d(v
j
).
Given the degree sequence D = {d
1
, d
2
, . . . , d
m
}, an extremal tree T that achieves the maximum ABC
index among the trees with degree sequence D can be constructed as follows:
(i) If d
m
≥ m −1, then by Corollary 3.2, the vertices with degrees respectively d
1
, d
2
, . . . , d
m−1
are all in L
1
,
and thus we construct an extremal tree T by rooting at vertex u with d
m
children with degrees d
1
, d
2
, . . . , d
m−1
and 1, . . . , 1
d
m
−m+1 times
.
(ii) Suppose that d
m
≤ m − 2.
(a) For the extremal tree T, by Corollary 3.2, the vertices in L
1
take some largest degrees and they
are adjacent to the vertices in L
2
with some smallest degrees. We construct some subtrees that contain
vertices in L
0
, L
1
and L
2
first. We produce subtree T
1
: rooted at vertex u
1
with d
m
− 1 children with degrees
d
1
, d
2
, . . . , d
d
m
−1
, where u
1
∈ L
2
, d
T
(u
1
) = d
m
, and the children of u
1
are all in L
1
. Removing T
1
except the
root u
1
from T results in a new tree S
1
with degree sequence D
1
= {d
d
m
, d
d
m
+1
, . . . , d
m−1
}. By Lemma 3.1 and
Corollary 3.2, S
1
is a tree with maximum ABC index among the trees with the degree sequence D
1
. Then
do the same to S
1
to get T
2
and S
2
, and then T
3
and S
3
, and so on, until S
k
satisfies the condition of (i).
(b) For i = k , k − 1, . . . , 1, the remaining is to identify u
i
with which pendent vertex of S
i
. Let v
i
be the
pendent vertex in S
i
with which u
i
is identified, and let w
i
be the unique neighbor of v
i
in S
i
. Since T is a
tree of degree sequence D with maximum ABC index, we need to maximize
ABC(T) = f(d
T
i
(u
i
) + 1, d
S
i
(w
i
)) + F,
where F is a constant independent of the pendent vertex of S
i
that we identify u
i
with. Note that d
T
i
(u
i
)+1 ≥ 2.
By Lemma 2.1, we need to minimize d
S
i
(w
i
).
Hence, we construct T as: identifying u
i
with a pendent vertex v
i
in S
i
, where w
i
is the unique neighbor
of v
i
in S
i
, such that w
i
∈ L
1
(S
i
) and d
S
i
(w
i
) = min{d
S
i
(x) : x ∈ L
1
(S
i
)}.
For an example, consider the degree sequence {4, 4, 3, 3, 3, 2, 2}. First, by (ii) a, we have the subtree
T
1
and new degree sequence D
1
= {4, 3, 3, 3, 2}, and similarly, the tree T
2
and still new degree sequence
D
2
= {3, 3, 3}. It is easily seen that D
2
satisfies the condition of (i), and thus we have S
2
. There are three
vertices in L
1
(S
2
) with degree three, two of which are symmetric in S
2
, and then by (ii) b, we have two types
of S
1
by identifying u
2
of T
2
and a pendent vertex of S
2
. Similarly, by identifying u
1
of T
1
and a pendent
R. Xing, B. Zhou / Filomat 26:4 (2012), 683–688 686
vertex of S
1
, we have three extremal trees (of fixed degree sequence {4, 4, 3, 3, 3, 2, 2}) with maximum ABC
index, see Fig. 1.
s
u
1
s
@
@
@s s s
T
1
From degrees 4 and 2
D
1
= {4, 3, 3, 3, 2}
s
u
2
s
@
@
@s s s
T
2
From degrees 4 and 2
D
2
= {3, 3, 3}
s
u
@
@
@s s s
B
B
B
s s
B
B
B
s s
S
2
From degrees 3, 3 and 3
su (w
2
)
@
@
@s s su
2
(v
2
)
B
B
B
s s
B
B
B
s s s
J
J
J
J
s s s
su
@
@
@s
w
2
s s
B
B
B
s
u
2
(v
2
)
s
B
B
B
s s
s
J
J
J
J
s s s
Attaching subtree T
2
to S
2
to get two types of S
1
su
@
@
@s
w
1
s s
B
B
B
s s
B
B
B
s s
u
2
s
J
J
J
J
s s s
u
1
(v
1
)
s
J
J
J
J
s s s
T
su
@
@
@s s sw
1
B
B
B
su
2
s
B
B
B
s su
1
(v
1
)
s
J
J
J
J
s s s
s
J
J
J
J
s s s
T
′
su
@
@
@s sw
1
s
B
B
B
s s
B
B
B
s
u
2
s
u
1
(v
1
)
s
J
J
J
J
s s s
J
J
J
J
s
J
J
J
J
s s s
T
′′
Attaching subtree T
1
to S
1
to get three extremal trees T, T
′
and T
′′
Fig. 1. The procedure to construct extremal trees of degree sequence {4, 4, 3, 3, 3, 2, 2}
with maximum ABC index.
Compared with the result in [12], an extremal tree T that achieves the maximum ABC index is just
the tree that achieves the maximum (general) Randi
´
c index for α < 0 among the trees with fixed degree
sequence.
4. Lower bound for the ABC index of trees with fixed degree sequence
In this section, we characterize the extremal trees with minimum ABC index among the trees with fixed
degree sequence, and provide an algorithm to construct such trees.
R. Xing, B. Zhou / Filomat 26:4 (2012), 683–688 687
Lemma 4.1. Let T be a tree with minimum ABC index among the trees with fixed degree sequence. Let P = v
1
v
2
. . . v
t
be a path in T, where t ≥ 4 and d(v
1
) < d(v
t
). Then d(v
2
) ≤ d(v
t−1
).
Proof. Suppose that d(v
2
) > d(v
t−1
). Let T
′
= T − {v
1
v
2
, v
t−1
v
t
} + {v
1
v
t−1
, v
2
v
t
}. Obviously, T
′
has the same
degree sequence as T. Since d(v
1
) < d(v
t
), we know by Lemma 2.2 that the function 1
d(v
1
),d(v
t
)
(x) is increasing
in x, and then
ABC(T) − ABC(T
′
) = f (d(v
1
), d(v
2
)) + f (d(v
t−1
), d(v
t
)) − f (d(v
1
), d(v
t−1
)) − f (d(v
2
), d(v
t
))
=
f (d(v
1
), d(v
2
)) − f (d(v
2
), d(v
t
))
−
f (d(v
1
), d(v
t−1
)) − f (d(v
t−1
), d(v
t
))
= 1
d(v
1
),d(v
t
)
(d(v
2
)) − 1
d(v
1
),d(v
t
)
(d(v
t−1
)) > 0,
which is a contradiction.
By Lemma 4.1, we have the following corollaries, as in [10].
Corollary 4.2. Let T be a tree with minimum ABC index among the trees with fixed degree sequence. Then there is
no path P = v
1
v
2
. . . v
t
in T with t ≥ 3 such that d(v
1
), d(v
t
) > d(v
i
) for some 2 ≤ i ≤ t − 1.
Corollary 4.3. Let T be a tree with minimum ABC index among the trees with fixed degree sequence. For every
positive integer d, the vertices with degrees at least d induce a subtree of T.
Corollary 4.4. Let T be a tree with minimum ABC index among the trees with fixed degree sequence. Then there are
no two non-adjacent edges v
1
v
2
and v
3
v
4
such that d(v
1
) < d(v
3
) ≤ d(v
4
) < d(v
2
).
By Corollary 4.3, the degrees of vertices in L
i
are no more than the degrees of vertices in L
i+1
for all
i = 0, 1, 2, . . . . Thus the vertices of larger degrees have farther distances from L
0
than the vertices of smaller
degrees.
Given the degree sequence D = {d
1
, d
2
, . . . , d
m
}, let T be a tree with minimum ABC index among the
trees with fixed degree sequence. If m = 1, then d
1
= |V(T)| − 1, and thus T is the star. Suppose that
m ≥ 2. Delorme et al. [10] discovered that the properties of extremal trees with maximum (general) Randi
´
c
index for α = 1 are the same as the features of Kruskal’s classical algorithm for the minimum spanning tree
problem. Wang [12] generalized it to the greedy algorithm.
Now an extremal tree T who achieves the minimum ABC index among the trees with fixed degree
sequence D = {d
1
, d
2
, . . . , d
m
} can be constructed as:
(i) Label a vertex with the largest degree d
1
as v, which is the root;
(ii) Label the neighbors of v as v
1
, v
2
, . . . , v
d
1
, such that d(v
1
) = d
2
≥ d(v
2
) = d
3
≥ ··· ≥ d(v
d
1
) = d
d
1
+1
;
(iii) Label the neighbors of v
1
except v as v
1,1
, v
1,2
, . . . , v
1,d
2
−1
such that d(v
1,1
) = d
d
1
+2
≥ d(v
1,2
) = d
d
1
+3
≥
··· ≥ d(v
1,d
2
−1
) = d
d
1
+d
2
, and do the same for the vertices v
2
, v
3
, . . . ;
(iv) Repeat (iii) for all the newly labeled vertices, and always start with the neighbors of the labeled
vertex with the largest degree whose neighbors are not labeled yet.
Now we give an example to construct extremal trees of degree sequence {4, 4, 4, 3, 3, 2, 2} with the
minimum ABC index, see Fig. 2.
s
v
T
T
T
T
H
H
H
H
H
Hs
v
1
s
v
2
s
v
3
s
v
4
T
T
T
s
v
1,1
s
v
1,2
s
A
A
As s s
B
B
B
s s
B
B
B
s s
s s
T
s
v
T
T
T
T
H
H
H
H
H
Hs
v
1
s
v
2
s
v
3
s
v
4
A
A
As
v
1,1
s s
A
A
As
v
2,1
s s
B
B
B
s s
B
B
B
s s
s s
T
′
Fig. 2. Two extremal trees T and T
′
of degree sequence {4, 4, 4, 3, 3, 2, 2} with minimum ABC index.
