Linear Algebra and its Applications 433 (2010) 1381–1387
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Linear Algebra and its Applications
journal homepage: www.elsevier.com/locate/laa
Note on unicyclic graphs with given number of pendent
vertices and minimal energy聻
Bofeng Huo a,b , Shengjin Ji a , Xueliang Li a,∗
a
b
Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, China
Department of Mathematics and Information Science, Qinghai Normal University, Xining 810008, China
A R T I C L E
I N F O
Article history:
Received 13 April 2010
Accepted 17 May 2010
Available online 15 June 2010
Submitted by R.A. Brualdi
Keywords:
Unicyclic graph
Minimal energy
Coulson integral formula
A B S T R A C T
For a simple graph G, the energy E (G) is defined as the sum of the
absolute values of all eigenvalues of its adjacency matrix. Let G (n, p)
denote the set of unicyclic graphs with n vertices and p pendent
vertices. In [H. Hua, M. Wang, Unicyclic graphs with given number
of pendent vertices and minimal energy, Linear Algebra Appl. 426
(2007) 478–489], Hua and Wang discussed the graphs that have
minimal energy in G (n, p), and determined the minimal-energy
graphs among almost all different cases of n and p. In their work,
certain case of the values was left as an open problem in which the
minimal-energy species have to be chosen in two candidate graphs,
but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials. This paper aims at
solving the problem completely, by using the well-known Coulson
integral formula.
© 2010 Elsevier Inc. All rights reserved.
1. Introduction
For a given simple graph G of order n, denote by A(G) the adjacency matrix of G. The characteristic
polynomial of A(G)
φ(G; x) = det(xI − A(G)) = xn + a1 xn−1 + · · · + an ,
is usually called the characteristic polynomial of G.
聻
Supported by NSFC No.10831001, PCSIRT and the “973" program.
∗ Corresponding author.
E-mail addresses: [email protected] (B. Hou), [email protected] (S. Ji), [email protected]
(X. Li).
0024-3795/$ - see front matter © 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.laa.2010.05.017
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B. Huo et al. / Linear Algebra and its Applications 433 (2010) 1381–1387
,p
,p
Fig. 1. (a) The graph Sn , (b) the graph Rn .
The roots of the characteristic polynomial of G, denoted by λ1 , λ2 , . . . , λn , are called the eigenvalues
of G. The energy of a graph G is then defined as
E (G)
=
n
i =1
|λi |.
For the calculation of energy, Coulson [2] deduced the following formula
1 +∞
ixφ (G, ix)
E (G) =
n−
dx,
π −∞
φ(G, ix)
(1)
√
−1. It can be derived into the following handy formula
where n is the number of vertices of G and i =
[6,5]
1 +∞ 1
log |xn φ(G; i/x)|dx.
E (G) =
π −∞ x2
Moreover, Gutman and Polansky [6] converted (1) into an explicit formula as follows:
⎞2
⎛ n
⎞2 ⎤
⎡⎛ n
2
+∞
2
1
1
⎟
⎜
⎟ ⎥
⎢⎜ E (G) =
log ⎣⎝
(−1)k a2k x2k ⎠ + ⎝ (−1)k a2k+1 x2k+1 ⎠ ⎦ dx,
2π −∞ x2
k =0
k =0
where a1 , a2 , . . . , an are the coefficients of the characteristic polynomial of G.
From the above one can see that if G1 and G2 are two graphs with the same number of vertices,
it is clear that E (G1 ) E (G2 ) if |ak (G1 )| |ak (G2 )| for all k. Generally, this method is quite effective
and has been utilized to figure out bipartite graphs [16,3,4,11,18,21,24], unicyclic graphs [1,7,9,10,12,
19,22,17,14], bicyclic graphs [8,15,20,23], etc., respectively, with the extremal (minimal or maximal)
energy. In the discussion of graphs with extremal energy, however, we frequently meet such a case in
which the extremal graph has to be chosen in two or more candidate graphs, and cannot be determined
completely by just comparing of the corresponding coefficients of their characteristic polynomials. We
have studied this kind of problem and solved one of them, see [13]. This paper will solve another for
such kind of problem. Our method is often effective on comparing of the energies of two graphs that
have small diameters.
The following lemma is a well-known result due to Gutman [5], which will be used in the sequel.
Lemma 1.1. If G1 and G2 are two graphs with the same number of vertices, then
1 +∞
φ(G1 ; ix)
E (G1 ) − E (G2 ) =
log
dx.
π −∞
φ(G2 ; ix)
2. Main results
We use G (n, p) to denote the set of unicyclic graphs with n vertices and p pendent vertices. Let
Sn (n + 1) be the graph obtained by identifying the center of the star Sn−+1 with any vertex of a
cycle C . Let Pn (n + 1) be the graph obtained by attaching one pendent vertex of the path Pn−+1
B. Huo et al. / Linear Algebra and its Applications 433 (2010) 1381–1387
4,p
1383
4,p
Fig. 2. (a) The graph Sp+5 , (b) the graph Rp+5 .
,p
with any vertex of the cycle C . Denote by Sn (n + p + 1) the graph obtained by attaching one
. By R,p (n + p + 1) we denote
pendent vertex of the path Pn−−p+1 to one pendent vertex of S+
n
p
the graph obtained by attaching p pendent edges to the pendent vertex of Pn−p . Hua and Wang [12]
planed to characterize graphs that have minimal energy among all graphs in G (n, p). They almost
completely solved this problem except for the case n = p + 5 for which the problem is reduced to
4,p
4,p
finding the minimal energy species among two graphs: Sp+5 and Rp+5 (see Figs. 1 and 2). In this paper,
4,p
we solve this problem and obtain that the minimal-energy graph in the case n = p + 5 is Rp+5 .
For the minimal energy unicyclic graphs with prescribed pendent vertices, Hua and Wang [12]
obtained the following result.
Theorem 2.1. Let 1 p n − 3. Then we have
3,p
4,p
(a) For p = n − 3, Sn has the minimal energy among all graphs in G (n, p); For p = n − 4, Sn has the
minimal energy among all graphs in G (n, p).
4,p
4,p
(b) For p = 1, Sn (= Rn ) has the minimal energy among all graphs in G (n, p).
4,p
(c) For p = n − 5 and 1 n 869, Rn has the minimal energy among all graphs in G (n, p), while for
4,p
4,p
p = n − 5 and n 870, Rn or Sn has the minimal energy among all graphs in G (n, p).
4,p
(d) For 2 p n − 6, Sn has the minimal energy among all graphs in G (n, p).
It is obvious that both of the two graphs have diameter 4. Their characteristic polynomials can be
expressed as follows.
Lemma 2.2 [12].
φ(Sp+5 ; x) = xp−1 (x6 − (p + 5)x4 + 3(p + 1)x2 − 2(p − 1));
4,p
φ(Rp+5 ; x) = xp+1 (x4 − (p + 5)x2 + (4p + 2)).
4,p
Before comparing of the energies of the two graphs, we recall two useful inequalities from [13].
Lemma 2.3. For any real number X
X
1+X
> −1, we have
log(1 + X ) X .
Lemma 2.4. Let A be a positive real number, B and C are non-negative. Then X
(2)
=
B −C
A +C
Now we give the main result of our paper.
Theorem 2.5. For p
4,p
= n − 5, Rn has the minimal energy among all graphs in G (n, p).
> −1.
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B. Huo et al. / Linear Algebra and its Applications 433 (2010) 1381–1387
4,p
4,p
Proof. Clearly, the common comparing method is invalid for Rp+5 and Sp+5 . That is, it cannot be determined completely by just comparing corresponding coefficients of their characteristic polynomials.
We use the Coulson integral formula to compare the energies of the two graphs. By Lemmas 1.1 and
2.2, it is easy to get that
1 +∞
x6 + (p + 5)x4 + (4p + 2)x2
4,p
4,p
log
dx.
(3)
E (Rp+5 ) − E (Sp+5 ) =
π −∞
x6 + (p + 5)x4 + (3p + 3)x2 + (2p − 2)
C
We denote by f (x, p) the integrand in Eq.(3). By letting A = x6 + (p + 5)x4 , B = (4p + 2)x2 and
= (3p + 3)x2 + (2p − 2), we can express f (x, p) as
B−C
A+B
= log 1 +
,
f (x, p) = log
A+C
A+C
i.e.,
(p − 1)x2 − (2p − 2)
.
f (x, p) = log 1 +
x6 + (p + 5)x4 + (3p + 3)x2 + (2p − 2)
Obviously, for p 1, x =
/ 0 we have A > 0, B 0 and C 0. Now let X
2.3 and 2.4, we get that for all x ∈ R, x =
/ 0 and any integer p 1,
f (x, p) x6
=
B −C
.
A +C
Then from Lemmas
(p − 1)x2 − (2p − 2)
+ (p + 5)x4 + (3p + 3)x2 + (2p − 2)
and
f (x, p) (p − 1)x2 − (2p − 2)
.
x6 + (p + 5)x4 + (4p + 2)x2
It follows that
f (x, p) x6
√
(p − 1)x2 − (2p − 2)
0, if 0 < |x| 2
4
2
+ (p + 5)x + (4p + 2)x
(4)
and
√
(p − 1)x2 − (2p − 2)
> 0, if |x| > 2.
x6 + (p + 5)x4 + (3p + 3)x2 + (2p − 2)
/ 0, and
Notice that the function sequence {f (x, p)} is convergent if x =
f (x, p) lim f (x, p)
p→+∞
Let ϕ(x)
= log
(5)
+ 4x2
.
x4 + 3x2 + 2
x4
= log x4x++3x4x2 +2 denote the limit of {f (x, p)}. For x =
/ 0, we have
4
2
+ 4x2
x6 + (p + 5)x4 + (4p + 2)x2
−
log
x4 + 3x2 + 2
x6 + (p + 5)x4 + (3p + 3)x2 + (2p − 2)
8
6
x + (p + 9)x + (7p + 23)x4 + (14p + 10)x2 + (8p − 8)
= log 8
.
x + (p + 8)x6 + (7p + 19)x4 + (14p + 16)x2 + (8p + 4)
x4
ϕ(x) − f (x, p) = log
= x8 + (p + 8)x6 + (7p + 19)x4 + (14p + 10)x2 + (8p − 8), B = x6 + 4x4 and C =
+B
C
6x + 12, we can express the above ϕ(x) − f (x, p) as ϕ(x) − f (x, p) = log AA+
= log 1 + AB−
C
+C , i.e.,
By letting A
2
ϕ(x) − f (x, p) = log 1 +
+ 4x4 − 6x2 − 12
.
8
6
x + (p + 8)x + (7p + 19)x4 + (14p + 16)x2 + (8p + 4)
x6
For x =
/ 0, it is easy to see that A > 0, B 0 and C 0. From Lemmas 2.3 and 2.4, for any x
and any positive integer p, we have
=
/ 0, x ∈ R
B. Huo et al. / Linear Algebra and its Applications 433 (2010) 1381–1387
ϕ(x) − f (x, p) + 4x4 − 6x2 − 12
x8 + (p + 8)x6 + (7p + 19)x4 + (14p + 16)x2 + (8p + 4)
ϕ(x) − f (x, p) + 4x4 − 6x2 − 12
,
x8 + (p + 9)x6 + (7p + 23)x4 + (14p + 10)x2 + (8p − 8)
1385
x6
and
x6
B−C
= x6 + 4x4 − 6x2 − 12 = (x2 − 2)(x4 + 6x2 + 6). It follows that
√
ϕ(x) < f (x, p), if 0 < |x| < 2
and
ϕ(x) > f (x, p), if |x| >
Similarly, for x
√
2.
(6)
(7)
=
/ 0, we have
f (x, p + 1) − f (x, p)
= log
x6 + (p + 6)x4 + (4p + 6)x2
6 )x 4
− log
x6 + (p + 5)x4 + (4p + 2)x2
+ (p +
+ (3p +
+ 2p
+ (p + 5)x4 + (3p + 3)x2 + (2p − 2)
8
2
6
x + (2p + 11)x + (p + 18p + 39)x + (7p2 + 49p + 46)x4 + (14p2 + 40p + 6)x2 + (8p2 + 4p − 12)
= log
x10 + (2p + 11)x8 + (p2 + 18p + 38)x6 + (7p2 + 49p + 42)x4 + (14p2 + 40p + 12)x2 + (8p2 + 4p)
x6
6 )x 2
x6
10
By letting A = x10 + (2p + 11)x8 + (p2 + 18p + 38)x6 + (7p2 + 49p + 42)x4 + (14p2 + 40p +
6)x2 + (8p2 + 4p − 12), B = x6 + 4x4 and C = 6x2 + 12, we can express the above f (x, p + 1) −
f (x, p) as
B−C
A+B
f (x, p + 1) − f (x, p) = log
= log 1 +
,
A+C
A+C
i.e.,
f (x, p + 1) − f (x, p)
= log 1 +
x6 + 4x4 − 6x2 − 12
.
x10 + (2p + 11)x8 + (p2 + 18p + 38)x6 + (7p2 + 49p + 42)x4 + (14p2 + 40p + 12)x2 + (8p2 + 4p)
We also have
f (x, p + 1) − f (x, p)
x6 + 4x4 − 6x2 − 12
x10
+ (2p +
11)x8
+
( p2
+
( p2
+ 18p +
38)x6
+ 18p +
39)x6
+ (7p2 + 49p + 42)x4 + (14p2 + 40p + 12)x2 + (8p2 + 4p)
and
f (x, p + 1) − f (x, p)
x6 + 4x4 − 6x2 − 12
x10
+ (2p +
11)x8
+ (7p2 + 49p + 46)x4 + (14p2 + 40p + 6)x2 + (8p2 + 4p − 12)
= x6 + 4x4 − 6x2 − 12 = (x2 − 2)(x4 + 6x2 + 6). It follows that
√
f (x, p + 1) < f (x, p), if 0 < |x| < 2
,
B−C
and
f (x, p + 1)
> f (x, p), if |x| >
√
2.
(8)
(9)
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B. Huo et al. / Linear Algebra and its Applications 433 (2010) 1381–1387
Combining Inequ.(4) with Inequ.(9), we have
ϕ(x) < f (x, p + 1) < f (x, p) 0, if 0 < |x| and
ϕ(x) > f (x, p + 1) > f (x, p) > 0 if |x| >
√
2
(10)
√
2.
(11)
Based on inequalities (10) and (11), for p > 2, we immediately get
+∞
√2
+∞
√2
+∞
f (x, p)dx =
f (x, p)dx + √
f (x, p)dx <
f (x, 2)dx + √
ϕ(x)dx
0
2
0
2
0
With the computer-aided calculation and Lemma 2.3, it is easy to verify that for any x
x2 − 2
x6 + 7x4 + 10x2
= log 1 + 6
f (x, 2) = log
x6 + 7x4 + 9x2 + 2
x + 7x4 + 9x2 + 2
<
x2
x6
+
7x4
−2
< 6
+ 9x2 + 2
x +
−2
+ 694
x2 +
75
x2
211 4
x
30
117
50
=
/ 0
.
Furthermore, we have
√2
0
<
=
f (x, 2)dx
√2
0
√2
x6
+
−2
+ 694
x2 +
75
x2
211 4
x
30
−2
dx
dx
+ 13
x2 + 13
10
√ √ √ 185 √
30
525 √
990 √
2 65
=−
15 arctan
−
3 arctan
6 +
130 arctan
0
x2
+
x2
117
50
27
5
x2
4674
9
.
= − 0.408541,
1102
15457
13
as well as
+∞
√
2
+∞
+ 4x2
dx
x4 + 3x2 + 2
2
√
√ √ 2
2
=−
π + 2 arctan 2 − 4 arctan
+π
ϕ(x)dx =
√
log
x4
2
2
.
= 0.368866.
4,p
4,p
Therefore, the difference between the energies of Rp+5 and Sp+5 must be less than certain neg +∞
4,p
4,p
f (x, p)dx. Combining with (c) of Theorem 2.1, we
ative number, since E (Rp+5 ) − E (Sp+5 ) = π2 0
4,p
conclude that E (Rp+5 ) has the minimal energy among all graphs in G (n, p).
References
[1] A. Chen, A. Chang, W.C. Shiu, Energy ordering of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 55 (2006)
95–102.
[2] C.A. Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules, Proc. Cambridge Phil. Soc. 36 (1940)
201–203.
[3] I. Gutman, Acylclic systems with extremal Hückel π -electron energy, Theor. Chim. Acta 45 (1977) 79–87.
[4] I. Gutman, Acylclic conjugated molecules, trees and their energies, J. Math. Chem. 1 (1987) 123–143.
B. Huo et al. / Linear Algebra and its Applications 433 (2010) 1381–1387
1387
[5] I. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann (Eds.), Algebraic
Combinatorics and Applications, Springer-Verlag, Berlin, 2001, pp. 196–211.
[6] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin, 1986.
[7] Y. Hou, Unicyclic graphs with minimal energy, J. Math. Chem. 29 (2001) 163–168.
[8] Y. Hou, Bicyclic graphs with minimal energy, Linear and Multilinear Algebra 49 (2001) 347–354.
[9] Y. Hou, I. Gutman, C. Woo, Unicyclic graphs with maximal energy, Linear Algebra Appl. 356 (2002) 27–36.
[10] H. Hua, On minimal energy of unicyclic graphs with prescribed girth and pendent vertices, MATCH Commun. Math. Comput.
Chem. 57 (2007) 351–361.
[11] H. Hua, Bipartite unicyclic graphs with large energy, MATCH Commun. Math. Comput. Chem. 58 (2007) 57–73.
[12] H. Hua, M. Wang, Unicyclic graphs with given number of pendent vertices and minimal energy, Linear Algebra Appl. 426
(2007) 478–489.
[13] B. Huo, X. Li, Y. Shi, L. Wang, Determining the conjugated trees with the third through the sixth minimal energies, MATCH
Commun. Math. Comput. Chem., in press.
[14] S. Li, X. Li, Z. Zhu, On minimal energy and Hosoya index of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 61
(2009) 325–339.
[15] X. Li, J. Zhang, On bicyclic graphs with maximal energy, Linear Algebra Appl. 427 (2007) 87–98.
[16] X. Li, J. Zhang, L. Wang, On bipartite graphs with minimal energy, Discrete Appl. Math. 157 (2009) 869–873.
[17] X. Li, J. Zhang, B. Zhou, On unicyclic conjugated molecules with minimal energies, J. Math. Chem. 42 (2007) 729–740.
[18] W. Lin, X. Guo, H. Li, On the extremal energies of trees with a given maximum degree, MATCH Commun. Math. Comput.
Chem. 54 (2005) 363–378.
[19] Y. Liu, Some results on energy of unicyclic graphswith n vertices, J. Math Chem. 47 (2010) 1–10.
[20] Z. Liu, B. Zhou, Minimal energies of bipartite bicyclic graphs, MATCH Commun. Math. Comput. Chem. 59 (2008) 381–396.
[21] H. Shan, J. Shao, S. Li, X. Li, On a conjecture on the tree with fourth greatest energy, MATCH Commun. Math. Comput. Chem.
64 (2009) 181–188.
[22] W. Wang, A. Chang, L. Zhang, D. Lu, Unicyclic Hückel molecular graphs with minimal energy, J. Math. Chem. 39 (2006)
231–241.
[23] Y. Yang, B. Zhou, Minimal energy of bicyclic graphs of a given diameter, MATCH Commun. Math. Comput. Chem. 59 (2008)
321–342.
[24] F. Zhang, Z. Lai, Three theorems of comparison of trees by their energy, Science Exploration 3 (1983) 12–19.