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Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs II
Accepted Manuscript Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs II
Lu Qiao, Shenggui Zhang, Jing Li
PII: DOI: Reference:
S0022-247X(16)30852-6 http://dx.doi.org/10.1016/j.jmaa.2016.12.056 YJMAA 20996
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
6 June 2016
Please cite this article in press as: L. Qiao et al., Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs II, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2016.12.056
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Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs II∗ Lu Qiao, Shenggui Zhang†, Jing Li Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, P.R. China
Abstract Le G be a graph of order n and λ1 ≥ λ2 ≥ · · · ≥ λn the eigenvalues of G. The n energy of G is defined as E(G) = k=1 |λk |. A well-known result on the energy of graphs is the Coulson integral formula which gives a relationship between the energy and the characteristic polynomial of graphs. Let μ1 ≥ μ2 ≥ · · · ≥ μn = 0 be the Laplacian eigenvalues of G. The general Laplacian-energy-like invariant of G, denoted α by LELα (G), is defined as μk =0 μk when μ1 = 0, and 0 when μ1 = 0, where α is a real number. In this paper we give some Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs in the case that α is a rational number. Based on this result, we further give some Coulson-type integral formulas for the general energy and general Laplacian energy of graphs in the case that α is a rational number. We also show that our formulas hold when α is an irrational number with 0 < |α| < 1 and do not hold with |α| > 1 . Keywords: Graph energy; General Laplacian-energy-like invariant; Coulson integral formula Mathematics Subject Classification: 05C50, 15A18.
1
Introduction
Throughout the paper we only consider simple graphs. For terminology and notation not defined here, we refer the reader to Cvetkovi´c et al. [3]. ∗
Supported by NSFC (Nos. 11271300, 11201374 and 11571135) and the Fundamental Research Funds
for the Central Universities (3102014JCQ01073). † Corresponding author. E-mail addresses: [email protected] (L. Qiao), sgzhang@nwpu. edu.cn (S. Zhang), [email protected] (J. Li).
1
Let G be a graph of order n. The spectrum of G consists of the eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn of the adjacency matrix A(G) of G, which are called the eigenvalues of G. It is well known that λ1 = max{|λ1 |, . . . , |λn |}. The Laplacian matrix of G is the matrix L(G) = D(G) − A(G), where D(G) = diag(d1 , d2 , . . . , dn ) is the diagonal matrix of vertex degrees of G. The Laplacian eigenvalues of G are the eigenvalues of L(G), denoted by μ1 ≥ μ2 ≥ · · · ≥ μn . As we all know, L(G) is a positive semi-definite symmetric matrix and μn = 0. The energy E(G) of G is defined as the sum of the absolute values of the eigenvalues of G, which is an invariant related to total π-electron energy [10]. Many mathematicians and chemists have done lots of work in the field of the theory of graph energy (see [7]). In 1940, Coulson [2] obtained an important integral formula which makes it possible to calculate the energy of a graph without knowing its spectrum. For a graph G on n vertices, its energy is 1 E(G) = π
+∞ −∞
ixφA (G, ix) dx, n− φA (G, ix)
where φA (G, x) is the characteristic polynomial of A(G) (the characteristic polynomial of G). This formula is called the Coulson integral formula, and has many applications in the theory of graph energy (see [7]). Moreover, Gutman and Zhou [6] defined the Laplacian energy of G as n 2m , μk − LE(G) = n k=1
where n and m are the number of vertices and edges of G, respectively. At the same time, Liu and Liu [8] defined the Laplacian-energy-like invariant of G as LEL(G) =
n √
μk .
k=1
This invariant has many similar properties as the energy of graphs. For results and problems on these two invariants, we refer the reader to reference [5, 7]. In [11], Zhou studied the sum of powers of the Laplacian eigenvalues of graphs, which can be regarded as a generalization of the Laplacian-energy-like invariant and is called the general Laplacian-energy-like invariant of graphs in [9]. Definition 1. Let G be a graph of order n, μ1 ≥ μ2 ≥ · · · ≥ μn = 0 the Laplacian eigenvalues of G and α a real number. The general Laplacian-energy-like invariant of G, denoted by LELα (G), is defined as μk =0 μαk when μ1 = 0, and 0 when μ1 = 0.
2
Qiao et al. [9] obtained an integral formula for general Laplacian-energy-like invariant in the case that α = 1/p, p ∈ Z+ \{1}, gave an extension of the general Laplacian-energylike invariant of graphs to complex polynomials and obtained an integral formula for it. Theorem 1 ([9]). Let G be a graph of order n, φL (G, x) the characteristic polynomial of the Laplacian matrix L(G) of G, and α = 1/p with p ∈ Z+ \{1}. Then the general Laplacian-energy-like invariant of G can be given by the following integral formula p +∞ xp φL (G, −xp ) π LELα (G) = + n · sin dx. π 0 φL (G, −xp ) p Definition 2 ([9]). Let φ(z) =
n
ak z n−k = a0
k=0
n
(z − zk )
k=1
be a complex polynomial of degree n and α a real number. The general energy of φ(z), denoted by Eα (φ(z)), is defined as zk =0 |zk |α when there exists i0 ∈ {1, 2, . . . , n} such that zi0 = 0, and 0 when z1 = · · · = zn = 0. Theorem 2 ([9]). Let φ(z) be a monic polynomial of degree n, whose roots are all nonnegative real numbers, and α = 1/p with p ∈ Z+ \{1}. Then the general energy of φ(z) can be given by the following integral formula π p +∞ xp φ (−xp ) Eα (φ(z)) = + n · sin dx. p π 0 φ(−x ) p The two following concepts are regarded as generalizations of graph energy and Laplacian graph energy, respectively. Definition 3. Let G be a graph of order n, λ1 ≥ λ2 ≥ · · · ≥ λn the eigenvalues of G and α a real number. The general energy of G, denoted by Eα (G), is defined as λk =0 |λk |α when λ1 = 0, and 0 when λ1 = 0. Definition 4. Let G be a graph, μ1 ≥ μ2 ≥ · · · ≥ μn = 0 the Laplacian eigenvalues of G and α a real number. The general Laplacian energy of G is defined as 2m α LEα (G) = μk − n . μk = 2m n
In this paper, we obtain some Coulson-type integral formulas for the general Laplacianenergy-like invariant of graphs and the general energy of polynomials with α ∈ Q in Section 3. Before that, in Section 2, we give some preliminaries. In Sections 4 and 5, we present Coulson-type integral formulas for the general energy and general Laplacian energy of graphs with α ∈ Q, respectively. We also show that our formulas in Theorem 5 (1) and (4), Theorem 6 (1) and (4) and Theorem 7 (1) and (4) hold when α is an irrational number with 0 < |α| < 1 and do not hold with |α| > 1. 3
2
Preliminaries
We first introduce some basic concepts and results in complex analysis which will be used later. Let D be a bounded domain. The boundary of D is denoted by ∂D. We need the following simple lemma. The proofs are omitted here. Lemma 1. Let Sr be the arc z(θ) = a0 + reiθ , θ1 ≤ θ ≤ θ2 , where a0 and r > 0 are two real numbers. If f (z) is a continuous function on the arc Sr for all small r such that lim
max |reiθ f (a0 + reiθ ) − λ| = 0,
r→0+ θ∈[θ1 ,θ2 ]
then
lim
r→0+
Sr
f (z)dz = i(θ2 − θ1 )λ.
Suppose f (z) = 1. By Cauchy’s Theorem and Integral Formula, we get ⎧ ⎨ 1, if z0 ∈ int(∂D) ; dζ 1 f (z0 ) = = 2πi ∂D ζ − z0 ⎩ 0, if z ∈ ext(∂D), 0
where z0 ∈ int(∂D) and z0 ∈ ext(∂D) mean that z0 lies in the interior of ∂D and in the exterior of ∂D, respectively. n n Let φ(z) = ak z n−k = a0 (z − zk ) be a complex polynomial of degree n. By direct k=0
computing, we get
That is
k=1
n
n
k=1
k=1
zk zφ (z) z = =n+ . φ(z) z − zk z − zk n
zk zφ (z) −n= . φ(z) z − zk k=1
If z1 , z2 , . . . , zn ∈ int(∂D), then we have n n zφ (z) 1 zk 1 − n dz = dz = zk . 2πi ∂D φ(z) 2πi ∂D z − zk k=1
3
k=1
Coulson-type integral formulas for the general Laplacianenergy-like invariant of graphs with α ∈ Q
Suppose that φ(z) = (z − z1 )(z − z2 ) · · · (z − zn ). Then n
√ √ √ √ ( z − zk )(− z − zk ) φ( z)φ(− z) =
=
k=1 n
(z − zk2 )(−1) = (−1)n
k=1
n
k=1
4
(z − zk2 ).
Therefore, we have n
√ ϕ(z) = (−1) φ( z)φ(− z) = (z − zk2 ).
√
n
k=1
Thus, by Theorem 2 it is easy to get the following theorem. Theorem 3. Let φ(z) be a monic polynomial of degree n, whose roots are all non-negative real numbers, and α = 1. Then E1 (φ(z)) can be given by the following integral formula 2 +∞ x2 ϕ (−x2 ) + n dx, E1 (φ(z)) = π 0 ϕ(−x2 ) √ √ where ϕ(z) = (−1)n φ( z)φ(− z). Our main results in this paper are as follows. Theorem 4. Let G be a graph of order n with c (< n) components, φL (G, x) the characteristic polynomial of the Laplacian matrix L(G) of G, and α ∈ Q. Suppose that 1
1
ϕL (G, z) = ei(q−1)nπ φL (G, z q ) φL (G, z q e
−i 2π q
1
) · · · φL (G, z q e
−i
2(q−1)π q
). Then the general
Laplacian-energy-like invariant of G can be given as follows (1) If α = 1/p, p ∈ Z+ \{1}, then p LELα (G) = π
+∞ p x φL (G, −xp ) φL (G, −xp ) 0
π + n · sin dx. p
(2) If α = q, q ∈ Z+ , then 2 +∞ x2 ϕ (−x2 ) + n dx, LELα (G) = π 0 ϕ(−x2 ) √ √ where ϕ(z) = (−1)n ϕL (G, z)ϕL (G, − z). (3) If α = q/p, p, q ∈ Z+ \{1}, then π p +∞ xp ϕL (G, −xp ) + n · sin dx. LELα (G) = p π 0 ϕL (G, −x ) p (4) If α = −1/p, p ∈ Z+ \{1}, then π p +∞ −x−p φL (G, −x−p ) − c · sin dx. LELα (G) = −p π 0 φL (G, −x ) p (5) If α = −q, q ∈ Z+ , then LELα (G) = where ϕ(z) =
(−1)n ϕ
2 π
+∞
0
−x−2 ϕ (−x−2 ) − c dx, ϕ (−x−2 )
√ √ L (G, z)ϕL (G, − z).
(6) If α = −q/p, p, q ∈ Z+ \{1}, then π p +∞ −x−p ϕL (G, −x−p ) − c · sin dx. LELα (G) = −p π 0 ϕL (G, −x ) p 5
Proof. (1) This is just the result of Theorem 1. (2) Let μ1 ≥ μ2 ≥ · · · ≥ μn = 0 be the roots of φL (G, x). Then φL (G, x) = (x − μ1 )(x − μ2 ) · · · (x − μn ). Therefore, we have 1
1
ϕL (G, z) = ei(q−1)nπ φL (G, z q )φL (G, z q e = ei(q−1)nπ = ei(q−1)nπ =
n
k=1
n
k=1 n
1
1
1
1
(z q − μk )(z q e
−i 2π q
−i 2π q
1
− μk ) · · · (z q e
(z q − μk )(z q − μk e
i 2π q
2(q−1)π q
1
−i
−i
2(q−1)π q
) · · · φL (G, z q e
1
) · · · (z q − μk e
i
)
− μk )
2(q−1)π q
)e−i(q−1)π
k=1
(z − μqk ).
By Theorem 3, we obtain that LELα (G) =
n k=1
= where ϕ(z) = (−1)n ϕL (G,
2 π
μqk = E1 (ϕL (G, z))
+∞ 2 x ϕ (−x2 )
ϕ(−x2 )
0
+ n dx,
√ √ z)ϕL (G, − z).
(3) By Theorem 2, it is easy to obtain that LELα (G) =
n k=1
=
p π
q
μkp =
n
1
(μqk ) p = E 1 (ϕL (G, z)) p
k=1 +∞ p x ϕL (G, −xp ) ϕL (G, −xp ) 0
π + n · sin dx. p
(4) Suppose that μ1 ≥ μ2 ≥ · · · ≥ μn−c > μn−(c+1) = · · · = μn = 0 are the roots of n−c (x − μk ). Therefore, we obtain φL (G, x). Thus we can write φL (G, x) as φL (G, x) = xc k=1
that
1 c 1 1 ( p − μk ) ϕL (G, z) = z φL (G, p ) = z pn · p z z z n−c
pn
k=1
= (−1)n−c = (−1)n−c
= (−1)
n−c
= (−1)n−c
n−c
k=1 n−c
k=1 n−c
(z p μk − 1) μk (z p − μk
p−1
t=0 k=1 n−c
1 ) μk − 1 −i 2tπ p
(z − μk p e
(−1)p−1
p−1
t=0
k=1
6
1 p
(zμk e
)
i 2tπ p
− 1) .
Then, n−c p−1
zϕL (G, z) = ϕL (G, z) =
=
z
− p1 −i 2tπ p k=1 t=0 z − μ k e 1 − −i 2tπ p−1 n−c μk p e p 1+ − 1 −i 2tπ k=1 t=0 z − μk p e p − 1 −i 2tπ p−1 n−c μk p e p p(n − c) + − p1 −i 2tπ p k=1 t=0 z − μ k e
Thus we have
− 1 −i 2tπ p
n−c p−1
μk p e
k=1 t=0
z − μk p e
zϕL (G, z) − p(n − c) = ϕL (G, z)
.
− 1 −i 2tπ p
.
Suppose that Γ = ΓR ∪ L1 ∪ Sr ∪ L2 (see Figure 1) is a positively (i.e., counter−1
p clockwisely) oriented piecewise smooth Jordan curve, where R > max{μ1 , μ−1 n−c , μn−c },
−1
p π π iθ 0 < r < min{μn−c , μ−1 1 , μ1 }, ΓR is the curve {z(θ) = Re , − p ≤ θ ≤ p }, L1 is the line
{z(θ) = ρeiθ , r ≤ ρ ≤ R, θ =
π p },
π π p ≤ θ ≤ p }, and −1 −1 − p1 Then the points μ1 p , μ2 p , . . . , μn−c are − p1 −i 2tπ − p1 −i 2tπ − p1 −i 2tπ 0, μ1 e p , μ2 e p , . . . , μn−c e p ,
Sr is the the curve {z(θ) = reiθ , −
L2 is the line {z(θ) = ρeiθ , r ≤ ρ ≤ R, θ = − πp }. all in the interior of the curve Γ, and the points
t = 1, 2, . . . , p−1, are all in the exterior of the curve Γ. By Cauchy’s Theorem and Integral Formula, we get y
L1 ΓR Sr−
x L2
Figure 1: The curve Γ in Theorem 4 (4).
7
1 2πi
Γ
p−1 n−c zϕL (G, z) 1 − p(n − c) dz = ϕL (G, z) 2πi Γ k=1 t=0
n−c 1 = 2πi Γ k=1
=
n−c 1 2πi Γ k=1
=
n−c k=1
− p1
μk
z
z
− 1 −i 2tπ p
μk p e z−
− 1 −i 2tπ μk p e p
−1 μk p −1 − μk p −1 μk p −1 − μk p
dz
dz
dz
= LEL− 1 (G). p
Since the value of the integral 1 2πi
Γ
zϕL (G, z) − p(n − c) dz ϕL (G, z)
is independent of the actual values of R and r, it can be gotten that zϕL (G, z) 1 lim − p(n − c) dz LELα (G) = 2πi R→+∞ Γ ϕL (G, z) + r→0 zϕL (G, z) zϕL (G, z) 1 = lim − p(n − c) dz + − p(n − c) dz 2πi R→+∞ ΓR ϕL (G, z) ϕL (G, z) L1 r→0+ zϕL (G, z) zϕL (G, z) − p(n − c) dz + − p(n − c) dz , + ϕL (G, z) ϕL (G, z) Sr− L2 where Sr− is the same curve as Sr but has clockwise orientation. Suppose that z = ρ(cos θ + i sin θ), where ρ > 0. Then − p1 − p1 −i 2tπ − p1 −i 2tπ 2tπ 2tπ p p | |ρ(cos θ + i sin θ) − μ μ |z − μ e e k (cos p − i sin p )| k 1 − k = = z |z| ρ 2
≥
− p1
ρ2 + μkp − 2ρμk ρ
−1
|ρ − μk p | = . ρ
Thus we have 2tπ −1 p−1 n−c zμk p e−i p zϕL (G, z) z − p(n − c) ≤ − p1 −i 2tπ ϕL (G, z) p k=1 t=0 z − μ k e −1 2tπ p−1 μ p e−i p n−c k = − 1 −i 2tπ p μk p e k=1 t=0 1 − z ≤
p−1 n−c k=1 t=0
8
− p1
ρμk
−1
|ρ − μk p |
.
Obviously,
1
p−1 n−c
ρμkp
k=1 t=0
|ρ − μkp |
1
→ 0, for ρ → 0.
Then by Lemma 1 we get that zϕL (G, z) − p(n − c) dz → 0, for r → 0. ϕL (G, z) Sr− − 1 −i 2tπ p
Suppose that ωkt = μk p e
−1
. Then |ωk0 | = |ωk1 | = · · · = |ωk(p−1) | = μk p . We have
2tπ −1 p−1 μk p e−i p z 1 t=0 z − μ− p e−i 2tπ p k 2tπ 2tπ 2tπ 2tπ −1 −1 −1 −1 p−1 μk p e−i p (z p−1 + z p−2 μk p e−i p + · · · + z(μk p e−i p )p−2 + (μk p e−i p )p−1 ) = z z p − μ−1 t=0 k p−1 ω (z p−1 + z p−2 ω + · · · + zω p−2 + ω p−1 ) kt kt kt kt = z p − μ−1 z k t=0 p−1 p 2 ωkt kt p−1 ωkt + ωzkt + · · · + ωz p−2 + z p−1 . = −1 μk t=0 1 − zp It is easy to get that there exists Nk > 0, for k ∈ {1, 2, . . . , n − c}, such that |1 −
μ−1 k zp |
≥
1 2
for |z| > Nk . For any ε > 0, there exists Mk > 0, for k ∈ {1, 2, . . . , n − c}, such that p−1 p−1 p ω 2 ω ω ε kt kt kt + · · · + p−2 , + p−1 < 2n z z z t=0
for |z| > Mk . Note that
p−1 t=0
r = 0 unless r = p. Therefore, for any ε > 0, there exists ωkt
N = max{N1 , N2 , . . . , Nn−c , M1 , M2 , . . . , Mn−c } such that − p1 −i 2tπ p−1 p zϕL (G, z) n−c μ e k z − p(n − c) = z 1 2tπ − ϕL (G, z) k=1 t=0 z − μ p e−i p k p−1 p ωkt ωkt ω2 p−1 n−c ωkt + zkt + · · · + z p−2 + z p−1 = −1 μ k=1 t=0 1 − zkp p−1 p−1 n−c n−c p−1 p 2 ωkt ωkt ωkt ≤ 2 + · · · + p−2 + p−1 ωkt + 2 z z z k=1 t=0 n−c
<0+2
k=1
k=1 t=0
ε < ε, 2n
for |z| > N . By standard estimate, we obtain that, for any ε > 0, there exists N =
9
max{N1 , N2 , . . . , Nn−c , M1 , M2 , . . . , Mn−c } such that the integral zϕL (G, z) zϕL (G, z) 2πR − p(n − c) dz ≤ max − p(n − c) ϕL (G, z) p z∈ΓR ϕL (G, z) ΓR zϕL (G, z) 2π = max z − p(n − c) p z∈ΓR ϕL (G, z) 2π < ε, p for |z| > N . This implies that zϕL (G, z) − p(n − c) dz → 0, for |z| → +∞. ϕL (G, z) ΓR Therefore, we obtain that zϕL (G, z) zϕL (G, z) 1 lim − p(n − c) dz + − LELα (G) = 2πi R→+∞ ΓR ϕL (G, z) L1 ϕL (G, z) r→0+ zϕL (G, z) zϕL (G, z) − p(n − c) dz + − p(n − c) dz + Sr− ϕL (G, z) L2 ϕL (G, z) p(n − c) dz zϕL (G, z) zϕL (G, z) 1 = lim − p(n − c) dz + − 2πi R→+∞ L1 ϕL (G, z) L2 ϕL (G, z) r→0+ p(n − c) dz φ (G, z −p ) −p φL (G, z −p ) −p 1 pc − L dz + pc − dz lim pz pz = 2πi R→+∞ L1 φL (G, z −p ) φL (G, z −p ) L2 r→0+
1 = lim 2πi R→+∞
r→0+
R
pc −
r R
pc −
iπ
φL (G, (ρe p )−p ) i πp
φL (G, (ρe )−p )
φL (G, (ρe
−i πp −p ) )
iπ iπ p(ρe p )−p d(ρe p )
−i πp −p
−i πp
) ) d(ρe p(ρe −i π φL (G, (ρe p )−p ) r π φ (G, −ρ−p ) 1 −p i p = lim (−p)ρ pc − L e dρ 2πi R→+∞ R φL (G, −ρ−p ) + r→0 R π φL (G, −ρ−p ) −p −i p + (−p)ρ dρ pc − e φL (G, −ρ−p ) r R φ (G, −ρ−p ) −p π 1 π = (cos + i sin )dρ lim pρ − pc + L −p 2πi R→+∞ r φL (G, −ρ ) p p r→0+ R φ (G, −ρ−p ) π π −p )pρ ) + i sin(− ))dρ + pc + L (cos(− φL (G, −ρ−p ) p p r R −p φ (G, −ρ ) −p p π =− lim ρ c+ L · sin dρ π R→+∞ r φL (G, −ρ−p ) p r→0+ p +∞ −x−p φL (G, −x−p ) π = − c · sin dx. π 0 φL (G, −x−p ) p +
r
10
Note that the formula above also holds for the general energy Eα (φ(z)) of φ(z) whose roots are all nonnegative (here c is the multiplicity of 0 as root of φ(z)). (5) Clearly, we have that LEL−q (G) = E−1/2 (ϕ(z)) = where ϕ(z) = (−1)n ϕL (G,
+∞
2 π
0
√ √ z)ϕL (G, − z).
−x−2 ϕ (−x−2 ) − c dx, ϕ (−x−2 )
(6) It can be easy to get that LEL− q (G) = p
n k=1
=
p π
− pq
μk
=
n
(μqk )
− p1
= LEL− 1 (ϕL (G, x))
k=1 +∞ −p −x ϕL (G, −x−p ) ϕL (G, −x−p ) 0
p
π − c · sin dx. p
The proof is complete. Suppose that φ(z) is a monic polynomial whose roots are all non-negative real numbers. Similar to the proof of the above theorem, we can get the integral formulas for the general energy of φ(x) as follows. Theorem 5. Let φ(z) be a monic complex polynomial, c ∈ {0, 1, . . . , n − 1} the multi1
1
plicity of 0 as root of φ(z) and α ∈ Q. Suppose that ϕ(z) = ei(q−1)nπ φ(z q )φ(z q e 1 q
φ(z e
2(q−1)π −i q
). Then the general energy of φ(z) can be given as follows
(1) If α = 1/p, p ∈ Z+ \{1}, then p Eα (φ(z)) = π (2) If α = q, q ∈
Z+ ,
+∞ p x φ (−xp )
π + n · sin dx. p φ(−x ) p
0
then
2 Eα (φ(z)) = π √ √ where P (z) = (−1)n ϕ( z)ϕ(− z).
+∞ 2 x P (−x2 )
P (−x2 )
0
+ n dx,
(3) If α = q/p, p, q ∈ Z+ \{1}, then π p +∞ xp ϕ (−xp ) + n · sin dx. Eα (φ(z)) = p π 0 ϕ(−x ) p (4) If α = −1/p, p ∈ Z+ \{1}, then π p +∞ −x−p φ (−x−p ) − c · sin dx. Eα (φ(z)) = −p π 0 φ(−x ) p (5) If α = −q, q ∈ Z+ , then Eα (φ(z)) = where P (z) = (−1)n ϕL (G,
2 π
+∞
0
−x−2 P (−x−2 ) − c dx, P (−x−2 )
√ √ z)ϕL (G, − z).
(6) If α = −q/p, p, q ∈ Z+ \{1}, then π p +∞ −x−p ϕ (−x−p ) − c · sin dx. Eα (φ(z)) = −p π 0 ϕ(−x ) p 11
−i 2π q
)···
4
Coulson-type integral formulas for the general energy of graphs with α ∈ Q
We define a new polynomial ϕA (G, z) = (−1)n φA (G,
√ √ z)φA (G, − z) in this section. Then
the roots of ϕA (G, z) are λ21 , λ22 , . . . , λ2n . Note that α λ2 2 = E α (ϕA (G, z)). |λk |α = Eα (G) = k 2 λk =0
λk =0
Thus, by Theorems 2 and 4 we have the following results. Theorem 6. Let G be a graph of order n, φA (G, x) the characteristic polynomial of the adjacency matrix A(G) of G, and c ∈ {0, 1, . . . , n − 1} is the multiplicity of 0 as root of √ √ φA (G, z). Suppose that ϕA (G, z) = (−1)n φA (G, z)φA (G, − z). Then the general energy of G can be given as follows (1) If α = 1/p, p ∈ Z+ , then 2p Eα (G) = π
(2) If α = q/p, p, q ∈ Z+ , then 2p Eα (G) = π
+∞ 2p x ϕA (G, −x2p ) ϕA (G, −x2p ) 0
1
+∞ 2p x ϕA (−x2p ) ϕA (−x2p ) 0 1
where ϕA (z) = ei(q−1)nπ ϕA (G, z q )ϕA (G, z q e
−i 2π q
π + n · sin dx. 2p
π + n · sin dx, 2p 1
) · · · ϕA (G, z q e
−i
2(q−1)π q
).
(3) If α = −1/p, p ∈ Z+ , then π 2p +∞ −x−2p ϕA (G, −x−2p ) − c · sin dx. Eα (G) = −2p π 0 ϕA (G, −x ) 2p (4) If α = −q/p, p, q ∈ Z+ , then π 2p +∞ −x−2p ϕA (−x−2p ) − c · sin dx, Eα (G) = π 0 ϕA (−x−2p ) 2p 1
1
where ϕA (z) = ei(q−1)nπ ϕA (G, z q )ϕA (G, z q e
5
−i 2π q
1
) · · · ϕA (G, z q e
−i
2(q−1)π q
).
Coulson-type integral formulas for the general Laplacian energy of graphs with α ∈ Q
√ Let G be a graph of order n and size m. Suppose that ϕL (G, z) = (−1)n φL (G, z + √ 2m 2m 2m 2 n )φL (G, − z + n ) in this section. Then the roots of ϕL (G, z) are (μ1 − n ) , (μ2 − 2m 2 n ) , . . . , (μn
−
2m 2 n ) .
Thus we get that α 2m α 2m 2 2 ) (μk − = E α2 (ϕL (G, z)). LEα (G) = μk − n = n 2m 2m μk =
n
μk =
n
By Theorems 2 and 4, we can get the following results. 12
Theorem 7. Let G be a graph of order n and size m, φL (G, x) the characteristic polynomial of the Laplacian matrix L(G) of G, and c ∈ {0, 1, . . . , n − 1} the multiplicity of 2m n √ √ 2m 2m n as roots of φL (G, z). Suppose that ϕL (G, z) = (−1) φL (G, z + n )φL (G, − z + n ). Then the general Laplacian energy of G can be given as follows (1) If α = 1/p, p ∈ Z+ , then 2p LEα (G) = π
+∞ 2p x ϕL (G, −x2p ) ϕL (G, −x2p ) 0
π + n · sin dx. 2p
(2) If α = q/p, p, q ∈ Z+ , then LEα (G) =
2p π 1
+∞ 2p x ϕL (−x2p ) ϕL (−x2p ) 0 1
where ϕL (z) = ei(q−1)nπ ϕL (G, z q )ϕL (G, z q e
−i 2π q
π + n · sin dx, 2p 1
) · · · ϕL (G, z q e
−i
2(q−1)π q
).
(3) If α = −1/p, p ∈ Z+ , then π 2p +∞ −x−2p ϕL (G, −x−2p ) LEα (G) = − c · sin dx. −2p π 0 ϕL (G, −x ) 2p (4) If α = −q/p, p, q ∈ Z+ , then π 2p +∞ −x−2p ϕL (−x−2p ) LEα (G) = − c · sin dx, −2p π 0 ϕL (−x ) 2p 1
1
where ϕL (z) = ei(q−1)nπ ϕL (G, z q )ϕL (G, z q e
6
−i 2π q
1
) · · · ϕL (G, z q e
−i
2(q−1)π q
).
Integral formulas for the case α ∈ R\Q
It is natural to ask that whether the formulas obtained above hold for the case that α is an irrational number. Now we consider this problem. Let 0 < α = 1/p < 1 be an irrational number. Then 1 < p ∈ R\Q. For a graph G, 1/p LELα (G) = μk =0 μk . The integral in Theorem 4 (1) is p +∞ xp φL (G, −xp ) π + n · sin dx p π 0 φL (G, −x ) p +∞ n p x p π = sin + n dx p π p 0 −x − μk k=1 n π +∞ μk p dx = sin π p 0 xp + μk k=1 μk p 1 π +∞ sin dx. = p π p 0 x + μk μk =0
By using the software Mathematica, we have 1 x 1 xa 1 dx = ; 1 + ; − ), F (1, 2 1 xa + b b a a b 13
where 2 F1 (a0 , a1 ; b0 ; x) =
∞
k=0
(a0 )k (a1 )k xk k! (b0 )k
is a hypergeometric function with a > 1, b > 0
and (z)k = z(z + 1)(z + 2) · · · (z + k − 1) (see [1]). Again, using the software Mathematica, we get
1 xa 1 1 1 lim x 2 F1 (1, ; 1 + ; − ) = b1/a Γ(1 + )Γ(1 − ) x→+∞ a a b a a
and
1 xa 1 lim x 2 F1 (1, ; 1 + ; − ) = 0, x→0 a a b
where Γ(x) is the Gamma function. Since Γ(1 − x)Γ(x) =
π sin πx
for 0 < x < 1 and Γ(1 + x) = xΓ(x), we have 1 π +∞ pμk sin dx π p 0 xp + μk pμk π x 1 xa +∞ 1 = sin · ; 1 + ; − ) F (1, 2 1 π p μk p p μk 0 π 1/p 1 p 1 = sin · μk Γ(1 + )Γ(1 − ) π p p p π 1 1 1 1/p p sin · Γ( )Γ(1 − ) =μk π p p p p 1 π π 1/p 1/p sin · = μk . =μk π p sin πp Therefore, p +∞ xp φL (G, −xp ) π + n · sin dx p π 0 φL (G, −x ) p +∞ μk p 1 π sin dx = π p 0 xp + μk μk =0
=
μk =0
1/p
μk
=LELα (G). Let −1 < α = −1/p < 0 be an irrational number. Then 1 < p ∈ R\Q. For a graph G
14
−1/p with c components, LELα (G) = μk =0 μk . The integral in Theorem 4 (4) is p +∞ −x−p φL (G, −x−p ) π − c · sin dx −p π 0 φL (G, −x ) p +∞ n −p −x p π = sin − c dx π p 0 −x−p − μk k=1 n−c 1 π +∞ p dx = sin π p 0 1 + μ k xp k=1 n−c π +∞ μ−1 p k dx = sin p + μ−1 π p x 0 k k=1 n−c pμ−1 1 π +∞ k sin = −1 dx p π p 0 x + μ k k=1 =
=
=
n−c pμ−1 k
k=1 n−c k=1 n−c k=1
π
sin
π x 1 xa +∞ 1 · −1 2 F1 (1, ; 1 + ; − −1 ) p μk p p μk 0
p π −1/p 1 1 sin · μk Γ(1 + )Γ(1 − ) π p p p −1/p
μk
=LELα (G). Different from the case that α is an irrational number with 0 < |α| < 1, the integral formulas in Theorem 4 for the case |α| > 1 do not hold when α is irrational. Note that π p +∞ xp φL (G, −xp ) + n · sin dx p π 0 φL (G, −x ) p μk p 1 π +∞ sin dx. = π p 0 xp + μk μk =0
Then it follows from that the improper integral +∞ 1 dx, (0 < p < 1, a > 0) p x +a 0 diverges that the integral p π
+∞ p x φL (G, −xp ) φL (G, −xp ) 0
π + n · sin dx p
diverges. It can also be shown that the formulas in Theorem 5 (1) and (4), Theorem 6 (1) and (4) and Theorem 7 (1) and (4) hold when α is an irrational number with 0 < |α| < 1 and do not hold with |α| > 1. We omit the details here. Acknowledgements. The authors are grateful to Professor Ivan Gutman for his valuable comments on an early version of this paper. 15
References [1] G.E. Andrews, R. Askey, R. Roy, Special funtions, Cambridge University Press, Cambridge, 1999. [2] C.A. Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules, Proc. Cambridge Phil. Soc. 36 (1940) 201–203. [3] D.M. Cvetkovi´c, M. Doob, H. Sachs, Spectra of Graphs — Theory and Application, Academic Press, New York, 1980. [4] T.W. Gamelin, Complex Analysis, Springer-Verlag, New York, 2001. [5] I. Gutman, X. Li (Eds), Energies of Graphs — Theory and Applications, Math. Chem. Monogr., vol.17, ISBN 978-86-6009-033-3, 2016, p.III+290, Kragujevac. [6] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37. [7] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. [8] J. Liu, B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 (2008) 355–372. [9] L. Qiao, S. Zhang, B. Ning, J. Li, Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs I, J. Math. Anal. Appl. 435 (2016) 1249– 1261. [10] K. Yates, H¨ uckel Molecular Orbital Theory, Academic Press, New York, 1978. [11] B. Zhou, On sum of powers of the Laplacian eigenvalues of graphs, Linear Algebra Appl. 429 (2008) 2239–2246.
16
Lu Qiao, Shenggui Zhang, Jing Li
PII: DOI: Reference:
S0022-247X(16)30852-6 http://dx.doi.org/10.1016/j.jmaa.2016.12.056 YJMAA 20996
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
6 June 2016
Please cite this article in press as: L. Qiao et al., Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs II, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2016.12.056
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Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs II∗ Lu Qiao, Shenggui Zhang†, Jing Li Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, P.R. China
Abstract Le G be a graph of order n and λ1 ≥ λ2 ≥ · · · ≥ λn the eigenvalues of G. The n energy of G is defined as E(G) = k=1 |λk |. A well-known result on the energy of graphs is the Coulson integral formula which gives a relationship between the energy and the characteristic polynomial of graphs. Let μ1 ≥ μ2 ≥ · · · ≥ μn = 0 be the Laplacian eigenvalues of G. The general Laplacian-energy-like invariant of G, denoted α by LELα (G), is defined as μk =0 μk when μ1 = 0, and 0 when μ1 = 0, where α is a real number. In this paper we give some Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs in the case that α is a rational number. Based on this result, we further give some Coulson-type integral formulas for the general energy and general Laplacian energy of graphs in the case that α is a rational number. We also show that our formulas hold when α is an irrational number with 0 < |α| < 1 and do not hold with |α| > 1 . Keywords: Graph energy; General Laplacian-energy-like invariant; Coulson integral formula Mathematics Subject Classification: 05C50, 15A18.
1
Introduction
Throughout the paper we only consider simple graphs. For terminology and notation not defined here, we refer the reader to Cvetkovi´c et al. [3]. ∗
Supported by NSFC (Nos. 11271300, 11201374 and 11571135) and the Fundamental Research Funds
for the Central Universities (3102014JCQ01073). † Corresponding author. E-mail addresses: [email protected] (L. Qiao), sgzhang@nwpu. edu.cn (S. Zhang), [email protected] (J. Li).
1
Let G be a graph of order n. The spectrum of G consists of the eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn of the adjacency matrix A(G) of G, which are called the eigenvalues of G. It is well known that λ1 = max{|λ1 |, . . . , |λn |}. The Laplacian matrix of G is the matrix L(G) = D(G) − A(G), where D(G) = diag(d1 , d2 , . . . , dn ) is the diagonal matrix of vertex degrees of G. The Laplacian eigenvalues of G are the eigenvalues of L(G), denoted by μ1 ≥ μ2 ≥ · · · ≥ μn . As we all know, L(G) is a positive semi-definite symmetric matrix and μn = 0. The energy E(G) of G is defined as the sum of the absolute values of the eigenvalues of G, which is an invariant related to total π-electron energy [10]. Many mathematicians and chemists have done lots of work in the field of the theory of graph energy (see [7]). In 1940, Coulson [2] obtained an important integral formula which makes it possible to calculate the energy of a graph without knowing its spectrum. For a graph G on n vertices, its energy is 1 E(G) = π
+∞ −∞
ixφA (G, ix) dx, n− φA (G, ix)
where φA (G, x) is the characteristic polynomial of A(G) (the characteristic polynomial of G). This formula is called the Coulson integral formula, and has many applications in the theory of graph energy (see [7]). Moreover, Gutman and Zhou [6] defined the Laplacian energy of G as n 2m , μk − LE(G) = n k=1
where n and m are the number of vertices and edges of G, respectively. At the same time, Liu and Liu [8] defined the Laplacian-energy-like invariant of G as LEL(G) =
n √
μk .
k=1
This invariant has many similar properties as the energy of graphs. For results and problems on these two invariants, we refer the reader to reference [5, 7]. In [11], Zhou studied the sum of powers of the Laplacian eigenvalues of graphs, which can be regarded as a generalization of the Laplacian-energy-like invariant and is called the general Laplacian-energy-like invariant of graphs in [9]. Definition 1. Let G be a graph of order n, μ1 ≥ μ2 ≥ · · · ≥ μn = 0 the Laplacian eigenvalues of G and α a real number. The general Laplacian-energy-like invariant of G, denoted by LELα (G), is defined as μk =0 μαk when μ1 = 0, and 0 when μ1 = 0.
2
Qiao et al. [9] obtained an integral formula for general Laplacian-energy-like invariant in the case that α = 1/p, p ∈ Z+ \{1}, gave an extension of the general Laplacian-energylike invariant of graphs to complex polynomials and obtained an integral formula for it. Theorem 1 ([9]). Let G be a graph of order n, φL (G, x) the characteristic polynomial of the Laplacian matrix L(G) of G, and α = 1/p with p ∈ Z+ \{1}. Then the general Laplacian-energy-like invariant of G can be given by the following integral formula p +∞ xp φL (G, −xp ) π LELα (G) = + n · sin dx. π 0 φL (G, −xp ) p Definition 2 ([9]). Let φ(z) =
n
ak z n−k = a0
k=0
n
(z − zk )
k=1
be a complex polynomial of degree n and α a real number. The general energy of φ(z), denoted by Eα (φ(z)), is defined as zk =0 |zk |α when there exists i0 ∈ {1, 2, . . . , n} such that zi0 = 0, and 0 when z1 = · · · = zn = 0. Theorem 2 ([9]). Let φ(z) be a monic polynomial of degree n, whose roots are all nonnegative real numbers, and α = 1/p with p ∈ Z+ \{1}. Then the general energy of φ(z) can be given by the following integral formula π p +∞ xp φ (−xp ) Eα (φ(z)) = + n · sin dx. p π 0 φ(−x ) p The two following concepts are regarded as generalizations of graph energy and Laplacian graph energy, respectively. Definition 3. Let G be a graph of order n, λ1 ≥ λ2 ≥ · · · ≥ λn the eigenvalues of G and α a real number. The general energy of G, denoted by Eα (G), is defined as λk =0 |λk |α when λ1 = 0, and 0 when λ1 = 0. Definition 4. Let G be a graph, μ1 ≥ μ2 ≥ · · · ≥ μn = 0 the Laplacian eigenvalues of G and α a real number. The general Laplacian energy of G is defined as 2m α LEα (G) = μk − n . μk = 2m n
In this paper, we obtain some Coulson-type integral formulas for the general Laplacianenergy-like invariant of graphs and the general energy of polynomials with α ∈ Q in Section 3. Before that, in Section 2, we give some preliminaries. In Sections 4 and 5, we present Coulson-type integral formulas for the general energy and general Laplacian energy of graphs with α ∈ Q, respectively. We also show that our formulas in Theorem 5 (1) and (4), Theorem 6 (1) and (4) and Theorem 7 (1) and (4) hold when α is an irrational number with 0 < |α| < 1 and do not hold with |α| > 1. 3
2
Preliminaries
We first introduce some basic concepts and results in complex analysis which will be used later. Let D be a bounded domain. The boundary of D is denoted by ∂D. We need the following simple lemma. The proofs are omitted here. Lemma 1. Let Sr be the arc z(θ) = a0 + reiθ , θ1 ≤ θ ≤ θ2 , where a0 and r > 0 are two real numbers. If f (z) is a continuous function on the arc Sr for all small r such that lim
max |reiθ f (a0 + reiθ ) − λ| = 0,
r→0+ θ∈[θ1 ,θ2 ]
then
lim
r→0+
Sr
f (z)dz = i(θ2 − θ1 )λ.
Suppose f (z) = 1. By Cauchy’s Theorem and Integral Formula, we get ⎧ ⎨ 1, if z0 ∈ int(∂D) ; dζ 1 f (z0 ) = = 2πi ∂D ζ − z0 ⎩ 0, if z ∈ ext(∂D), 0
where z0 ∈ int(∂D) and z0 ∈ ext(∂D) mean that z0 lies in the interior of ∂D and in the exterior of ∂D, respectively. n n Let φ(z) = ak z n−k = a0 (z − zk ) be a complex polynomial of degree n. By direct k=0
computing, we get
That is
k=1
n
n
k=1
k=1
zk zφ (z) z = =n+ . φ(z) z − zk z − zk n
zk zφ (z) −n= . φ(z) z − zk k=1
If z1 , z2 , . . . , zn ∈ int(∂D), then we have n n zφ (z) 1 zk 1 − n dz = dz = zk . 2πi ∂D φ(z) 2πi ∂D z − zk k=1
3
k=1
Coulson-type integral formulas for the general Laplacianenergy-like invariant of graphs with α ∈ Q
Suppose that φ(z) = (z − z1 )(z − z2 ) · · · (z − zn ). Then n
√ √ √ √ ( z − zk )(− z − zk ) φ( z)φ(− z) =
=
k=1 n
(z − zk2 )(−1) = (−1)n
k=1
n
k=1
4
(z − zk2 ).
Therefore, we have n
√ ϕ(z) = (−1) φ( z)φ(− z) = (z − zk2 ).
√
n
k=1
Thus, by Theorem 2 it is easy to get the following theorem. Theorem 3. Let φ(z) be a monic polynomial of degree n, whose roots are all non-negative real numbers, and α = 1. Then E1 (φ(z)) can be given by the following integral formula 2 +∞ x2 ϕ (−x2 ) + n dx, E1 (φ(z)) = π 0 ϕ(−x2 ) √ √ where ϕ(z) = (−1)n φ( z)φ(− z). Our main results in this paper are as follows. Theorem 4. Let G be a graph of order n with c (< n) components, φL (G, x) the characteristic polynomial of the Laplacian matrix L(G) of G, and α ∈ Q. Suppose that 1
1
ϕL (G, z) = ei(q−1)nπ φL (G, z q ) φL (G, z q e
−i 2π q
1
) · · · φL (G, z q e
−i
2(q−1)π q
). Then the general
Laplacian-energy-like invariant of G can be given as follows (1) If α = 1/p, p ∈ Z+ \{1}, then p LELα (G) = π
+∞ p x φL (G, −xp ) φL (G, −xp ) 0
π + n · sin dx. p
(2) If α = q, q ∈ Z+ , then 2 +∞ x2 ϕ (−x2 ) + n dx, LELα (G) = π 0 ϕ(−x2 ) √ √ where ϕ(z) = (−1)n ϕL (G, z)ϕL (G, − z). (3) If α = q/p, p, q ∈ Z+ \{1}, then π p +∞ xp ϕL (G, −xp ) + n · sin dx. LELα (G) = p π 0 ϕL (G, −x ) p (4) If α = −1/p, p ∈ Z+ \{1}, then π p +∞ −x−p φL (G, −x−p ) − c · sin dx. LELα (G) = −p π 0 φL (G, −x ) p (5) If α = −q, q ∈ Z+ , then LELα (G) = where ϕ(z) =
(−1)n ϕ
2 π
+∞
0
−x−2 ϕ (−x−2 ) − c dx, ϕ (−x−2 )
√ √ L (G, z)ϕL (G, − z).
(6) If α = −q/p, p, q ∈ Z+ \{1}, then π p +∞ −x−p ϕL (G, −x−p ) − c · sin dx. LELα (G) = −p π 0 ϕL (G, −x ) p 5
Proof. (1) This is just the result of Theorem 1. (2) Let μ1 ≥ μ2 ≥ · · · ≥ μn = 0 be the roots of φL (G, x). Then φL (G, x) = (x − μ1 )(x − μ2 ) · · · (x − μn ). Therefore, we have 1
1
ϕL (G, z) = ei(q−1)nπ φL (G, z q )φL (G, z q e = ei(q−1)nπ = ei(q−1)nπ =
n
k=1
n
k=1 n
1
1
1
1
(z q − μk )(z q e
−i 2π q
−i 2π q
1
− μk ) · · · (z q e
(z q − μk )(z q − μk e
i 2π q
2(q−1)π q
1
−i
−i
2(q−1)π q
) · · · φL (G, z q e
1
) · · · (z q − μk e
i
)
− μk )
2(q−1)π q
)e−i(q−1)π
k=1
(z − μqk ).
By Theorem 3, we obtain that LELα (G) =
n k=1
= where ϕ(z) = (−1)n ϕL (G,
2 π
μqk = E1 (ϕL (G, z))
+∞ 2 x ϕ (−x2 )
ϕ(−x2 )
0
+ n dx,
√ √ z)ϕL (G, − z).
(3) By Theorem 2, it is easy to obtain that LELα (G) =
n k=1
=
p π
q
μkp =
n
1
(μqk ) p = E 1 (ϕL (G, z)) p
k=1 +∞ p x ϕL (G, −xp ) ϕL (G, −xp ) 0
π + n · sin dx. p
(4) Suppose that μ1 ≥ μ2 ≥ · · · ≥ μn−c > μn−(c+1) = · · · = μn = 0 are the roots of n−c (x − μk ). Therefore, we obtain φL (G, x). Thus we can write φL (G, x) as φL (G, x) = xc k=1
that
1 c 1 1 ( p − μk ) ϕL (G, z) = z φL (G, p ) = z pn · p z z z n−c
pn
k=1
= (−1)n−c = (−1)n−c
= (−1)
n−c
= (−1)n−c
n−c
k=1 n−c
k=1 n−c
(z p μk − 1) μk (z p − μk
p−1
t=0 k=1 n−c
1 ) μk − 1 −i 2tπ p
(z − μk p e
(−1)p−1
p−1
t=0
k=1
6
1 p
(zμk e
)
i 2tπ p
− 1) .
Then, n−c p−1
zϕL (G, z) = ϕL (G, z) =
=
z
− p1 −i 2tπ p k=1 t=0 z − μ k e 1 − −i 2tπ p−1 n−c μk p e p 1+ − 1 −i 2tπ k=1 t=0 z − μk p e p − 1 −i 2tπ p−1 n−c μk p e p p(n − c) + − p1 −i 2tπ p k=1 t=0 z − μ k e
Thus we have
− 1 −i 2tπ p
n−c p−1
μk p e
k=1 t=0
z − μk p e
zϕL (G, z) − p(n − c) = ϕL (G, z)
.
− 1 −i 2tπ p
.
Suppose that Γ = ΓR ∪ L1 ∪ Sr ∪ L2 (see Figure 1) is a positively (i.e., counter−1
p clockwisely) oriented piecewise smooth Jordan curve, where R > max{μ1 , μ−1 n−c , μn−c },
−1
p π π iθ 0 < r < min{μn−c , μ−1 1 , μ1 }, ΓR is the curve {z(θ) = Re , − p ≤ θ ≤ p }, L1 is the line
{z(θ) = ρeiθ , r ≤ ρ ≤ R, θ =
π p },
π π p ≤ θ ≤ p }, and −1 −1 − p1 Then the points μ1 p , μ2 p , . . . , μn−c are − p1 −i 2tπ − p1 −i 2tπ − p1 −i 2tπ 0, μ1 e p , μ2 e p , . . . , μn−c e p ,
Sr is the the curve {z(θ) = reiθ , −
L2 is the line {z(θ) = ρeiθ , r ≤ ρ ≤ R, θ = − πp }. all in the interior of the curve Γ, and the points
t = 1, 2, . . . , p−1, are all in the exterior of the curve Γ. By Cauchy’s Theorem and Integral Formula, we get y
L1 ΓR Sr−
x L2
Figure 1: The curve Γ in Theorem 4 (4).
7
1 2πi
Γ
p−1 n−c zϕL (G, z) 1 − p(n − c) dz = ϕL (G, z) 2πi Γ k=1 t=0
n−c 1 = 2πi Γ k=1
=
n−c 1 2πi Γ k=1
=
n−c k=1
− p1
μk
z
z
− 1 −i 2tπ p
μk p e z−
− 1 −i 2tπ μk p e p
−1 μk p −1 − μk p −1 μk p −1 − μk p
dz
dz
dz
= LEL− 1 (G). p
Since the value of the integral 1 2πi
Γ
zϕL (G, z) − p(n − c) dz ϕL (G, z)
is independent of the actual values of R and r, it can be gotten that zϕL (G, z) 1 lim − p(n − c) dz LELα (G) = 2πi R→+∞ Γ ϕL (G, z) + r→0 zϕL (G, z) zϕL (G, z) 1 = lim − p(n − c) dz + − p(n − c) dz 2πi R→+∞ ΓR ϕL (G, z) ϕL (G, z) L1 r→0+ zϕL (G, z) zϕL (G, z) − p(n − c) dz + − p(n − c) dz , + ϕL (G, z) ϕL (G, z) Sr− L2 where Sr− is the same curve as Sr but has clockwise orientation. Suppose that z = ρ(cos θ + i sin θ), where ρ > 0. Then − p1 − p1 −i 2tπ − p1 −i 2tπ 2tπ 2tπ p p | |ρ(cos θ + i sin θ) − μ μ |z − μ e e k (cos p − i sin p )| k 1 − k = = z |z| ρ 2
≥
− p1
ρ2 + μkp − 2ρμk ρ
−1
|ρ − μk p | = . ρ
Thus we have 2tπ −1 p−1 n−c zμk p e−i p zϕL (G, z) z − p(n − c) ≤ − p1 −i 2tπ ϕL (G, z) p k=1 t=0 z − μ k e −1 2tπ p−1 μ p e−i p n−c k = − 1 −i 2tπ p μk p e k=1 t=0 1 − z ≤
p−1 n−c k=1 t=0
8
− p1
ρμk
−1
|ρ − μk p |
.
Obviously,
1
p−1 n−c
ρμkp
k=1 t=0
|ρ − μkp |
1
→ 0, for ρ → 0.
Then by Lemma 1 we get that zϕL (G, z) − p(n − c) dz → 0, for r → 0. ϕL (G, z) Sr− − 1 −i 2tπ p
Suppose that ωkt = μk p e
−1
. Then |ωk0 | = |ωk1 | = · · · = |ωk(p−1) | = μk p . We have
2tπ −1 p−1 μk p e−i p z 1 t=0 z − μ− p e−i 2tπ p k 2tπ 2tπ 2tπ 2tπ −1 −1 −1 −1 p−1 μk p e−i p (z p−1 + z p−2 μk p e−i p + · · · + z(μk p e−i p )p−2 + (μk p e−i p )p−1 ) = z z p − μ−1 t=0 k p−1 ω (z p−1 + z p−2 ω + · · · + zω p−2 + ω p−1 ) kt kt kt kt = z p − μ−1 z k t=0 p−1 p 2 ωkt kt p−1 ωkt + ωzkt + · · · + ωz p−2 + z p−1 . = −1 μk t=0 1 − zp It is easy to get that there exists Nk > 0, for k ∈ {1, 2, . . . , n − c}, such that |1 −
μ−1 k zp |
≥
1 2
for |z| > Nk . For any ε > 0, there exists Mk > 0, for k ∈ {1, 2, . . . , n − c}, such that p−1 p−1 p ω 2 ω ω ε kt kt kt + · · · + p−2 , + p−1 < 2n z z z t=0
for |z| > Mk . Note that
p−1 t=0
r = 0 unless r = p. Therefore, for any ε > 0, there exists ωkt
N = max{N1 , N2 , . . . , Nn−c , M1 , M2 , . . . , Mn−c } such that − p1 −i 2tπ p−1 p zϕL (G, z) n−c μ e k z − p(n − c) = z 1 2tπ − ϕL (G, z) k=1 t=0 z − μ p e−i p k p−1 p ωkt ωkt ω2 p−1 n−c ωkt + zkt + · · · + z p−2 + z p−1 = −1 μ k=1 t=0 1 − zkp p−1 p−1 n−c n−c p−1 p 2 ωkt ωkt ωkt ≤ 2 + · · · + p−2 + p−1 ωkt + 2 z z z k=1 t=0 n−c
<0+2
k=1
k=1 t=0
ε < ε, 2n
for |z| > N . By standard estimate, we obtain that, for any ε > 0, there exists N =
9
max{N1 , N2 , . . . , Nn−c , M1 , M2 , . . . , Mn−c } such that the integral zϕL (G, z) zϕL (G, z) 2πR − p(n − c) dz ≤ max − p(n − c) ϕL (G, z) p z∈ΓR ϕL (G, z) ΓR zϕL (G, z) 2π = max z − p(n − c) p z∈ΓR ϕL (G, z) 2π < ε, p for |z| > N . This implies that zϕL (G, z) − p(n − c) dz → 0, for |z| → +∞. ϕL (G, z) ΓR Therefore, we obtain that zϕL (G, z) zϕL (G, z) 1 lim − p(n − c) dz + − LELα (G) = 2πi R→+∞ ΓR ϕL (G, z) L1 ϕL (G, z) r→0+ zϕL (G, z) zϕL (G, z) − p(n − c) dz + − p(n − c) dz + Sr− ϕL (G, z) L2 ϕL (G, z) p(n − c) dz zϕL (G, z) zϕL (G, z) 1 = lim − p(n − c) dz + − 2πi R→+∞ L1 ϕL (G, z) L2 ϕL (G, z) r→0+ p(n − c) dz φ (G, z −p ) −p φL (G, z −p ) −p 1 pc − L dz + pc − dz lim pz pz = 2πi R→+∞ L1 φL (G, z −p ) φL (G, z −p ) L2 r→0+
1 = lim 2πi R→+∞
r→0+
R
pc −
r R
pc −
iπ
φL (G, (ρe p )−p ) i πp
φL (G, (ρe )−p )
φL (G, (ρe
−i πp −p ) )
iπ iπ p(ρe p )−p d(ρe p )
−i πp −p
−i πp
) ) d(ρe p(ρe −i π φL (G, (ρe p )−p ) r π φ (G, −ρ−p ) 1 −p i p = lim (−p)ρ pc − L e dρ 2πi R→+∞ R φL (G, −ρ−p ) + r→0 R π φL (G, −ρ−p ) −p −i p + (−p)ρ dρ pc − e φL (G, −ρ−p ) r R φ (G, −ρ−p ) −p π 1 π = (cos + i sin )dρ lim pρ − pc + L −p 2πi R→+∞ r φL (G, −ρ ) p p r→0+ R φ (G, −ρ−p ) π π −p )pρ ) + i sin(− ))dρ + pc + L (cos(− φL (G, −ρ−p ) p p r R −p φ (G, −ρ ) −p p π =− lim ρ c+ L · sin dρ π R→+∞ r φL (G, −ρ−p ) p r→0+ p +∞ −x−p φL (G, −x−p ) π = − c · sin dx. π 0 φL (G, −x−p ) p +
r
10
Note that the formula above also holds for the general energy Eα (φ(z)) of φ(z) whose roots are all nonnegative (here c is the multiplicity of 0 as root of φ(z)). (5) Clearly, we have that LEL−q (G) = E−1/2 (ϕ(z)) = where ϕ(z) = (−1)n ϕL (G,
+∞
2 π
0
√ √ z)ϕL (G, − z).
−x−2 ϕ (−x−2 ) − c dx, ϕ (−x−2 )
(6) It can be easy to get that LEL− q (G) = p
n k=1
=
p π
− pq
μk
=
n
(μqk )
− p1
= LEL− 1 (ϕL (G, x))
k=1 +∞ −p −x ϕL (G, −x−p ) ϕL (G, −x−p ) 0
p
π − c · sin dx. p
The proof is complete. Suppose that φ(z) is a monic polynomial whose roots are all non-negative real numbers. Similar to the proof of the above theorem, we can get the integral formulas for the general energy of φ(x) as follows. Theorem 5. Let φ(z) be a monic complex polynomial, c ∈ {0, 1, . . . , n − 1} the multi1
1
plicity of 0 as root of φ(z) and α ∈ Q. Suppose that ϕ(z) = ei(q−1)nπ φ(z q )φ(z q e 1 q
φ(z e
2(q−1)π −i q
). Then the general energy of φ(z) can be given as follows
(1) If α = 1/p, p ∈ Z+ \{1}, then p Eα (φ(z)) = π (2) If α = q, q ∈
Z+ ,
+∞ p x φ (−xp )
π + n · sin dx. p φ(−x ) p
0
then
2 Eα (φ(z)) = π √ √ where P (z) = (−1)n ϕ( z)ϕ(− z).
+∞ 2 x P (−x2 )
P (−x2 )
0
+ n dx,
(3) If α = q/p, p, q ∈ Z+ \{1}, then π p +∞ xp ϕ (−xp ) + n · sin dx. Eα (φ(z)) = p π 0 ϕ(−x ) p (4) If α = −1/p, p ∈ Z+ \{1}, then π p +∞ −x−p φ (−x−p ) − c · sin dx. Eα (φ(z)) = −p π 0 φ(−x ) p (5) If α = −q, q ∈ Z+ , then Eα (φ(z)) = where P (z) = (−1)n ϕL (G,
2 π
+∞
0
−x−2 P (−x−2 ) − c dx, P (−x−2 )
√ √ z)ϕL (G, − z).
(6) If α = −q/p, p, q ∈ Z+ \{1}, then π p +∞ −x−p ϕ (−x−p ) − c · sin dx. Eα (φ(z)) = −p π 0 ϕ(−x ) p 11
−i 2π q
)···
4
Coulson-type integral formulas for the general energy of graphs with α ∈ Q
We define a new polynomial ϕA (G, z) = (−1)n φA (G,
√ √ z)φA (G, − z) in this section. Then
the roots of ϕA (G, z) are λ21 , λ22 , . . . , λ2n . Note that α λ2 2 = E α (ϕA (G, z)). |λk |α = Eα (G) = k 2 λk =0
λk =0
Thus, by Theorems 2 and 4 we have the following results. Theorem 6. Let G be a graph of order n, φA (G, x) the characteristic polynomial of the adjacency matrix A(G) of G, and c ∈ {0, 1, . . . , n − 1} is the multiplicity of 0 as root of √ √ φA (G, z). Suppose that ϕA (G, z) = (−1)n φA (G, z)φA (G, − z). Then the general energy of G can be given as follows (1) If α = 1/p, p ∈ Z+ , then 2p Eα (G) = π
(2) If α = q/p, p, q ∈ Z+ , then 2p Eα (G) = π
+∞ 2p x ϕA (G, −x2p ) ϕA (G, −x2p ) 0
1
+∞ 2p x ϕA (−x2p ) ϕA (−x2p ) 0 1
where ϕA (z) = ei(q−1)nπ ϕA (G, z q )ϕA (G, z q e
−i 2π q
π + n · sin dx. 2p
π + n · sin dx, 2p 1
) · · · ϕA (G, z q e
−i
2(q−1)π q
).
(3) If α = −1/p, p ∈ Z+ , then π 2p +∞ −x−2p ϕA (G, −x−2p ) − c · sin dx. Eα (G) = −2p π 0 ϕA (G, −x ) 2p (4) If α = −q/p, p, q ∈ Z+ , then π 2p +∞ −x−2p ϕA (−x−2p ) − c · sin dx, Eα (G) = π 0 ϕA (−x−2p ) 2p 1
1
where ϕA (z) = ei(q−1)nπ ϕA (G, z q )ϕA (G, z q e
5
−i 2π q
1
) · · · ϕA (G, z q e
−i
2(q−1)π q
).
Coulson-type integral formulas for the general Laplacian energy of graphs with α ∈ Q
√ Let G be a graph of order n and size m. Suppose that ϕL (G, z) = (−1)n φL (G, z + √ 2m 2m 2m 2 n )φL (G, − z + n ) in this section. Then the roots of ϕL (G, z) are (μ1 − n ) , (μ2 − 2m 2 n ) , . . . , (μn
−
2m 2 n ) .
Thus we get that α 2m α 2m 2 2 ) (μk − = E α2 (ϕL (G, z)). LEα (G) = μk − n = n 2m 2m μk =
n
μk =
n
By Theorems 2 and 4, we can get the following results. 12
Theorem 7. Let G be a graph of order n and size m, φL (G, x) the characteristic polynomial of the Laplacian matrix L(G) of G, and c ∈ {0, 1, . . . , n − 1} the multiplicity of 2m n √ √ 2m 2m n as roots of φL (G, z). Suppose that ϕL (G, z) = (−1) φL (G, z + n )φL (G, − z + n ). Then the general Laplacian energy of G can be given as follows (1) If α = 1/p, p ∈ Z+ , then 2p LEα (G) = π
+∞ 2p x ϕL (G, −x2p ) ϕL (G, −x2p ) 0
π + n · sin dx. 2p
(2) If α = q/p, p, q ∈ Z+ , then LEα (G) =
2p π 1
+∞ 2p x ϕL (−x2p ) ϕL (−x2p ) 0 1
where ϕL (z) = ei(q−1)nπ ϕL (G, z q )ϕL (G, z q e
−i 2π q
π + n · sin dx, 2p 1
) · · · ϕL (G, z q e
−i
2(q−1)π q
).
(3) If α = −1/p, p ∈ Z+ , then π 2p +∞ −x−2p ϕL (G, −x−2p ) LEα (G) = − c · sin dx. −2p π 0 ϕL (G, −x ) 2p (4) If α = −q/p, p, q ∈ Z+ , then π 2p +∞ −x−2p ϕL (−x−2p ) LEα (G) = − c · sin dx, −2p π 0 ϕL (−x ) 2p 1
1
where ϕL (z) = ei(q−1)nπ ϕL (G, z q )ϕL (G, z q e
6
−i 2π q
1
) · · · ϕL (G, z q e
−i
2(q−1)π q
).
Integral formulas for the case α ∈ R\Q
It is natural to ask that whether the formulas obtained above hold for the case that α is an irrational number. Now we consider this problem. Let 0 < α = 1/p < 1 be an irrational number. Then 1 < p ∈ R\Q. For a graph G, 1/p LELα (G) = μk =0 μk . The integral in Theorem 4 (1) is p +∞ xp φL (G, −xp ) π + n · sin dx p π 0 φL (G, −x ) p +∞ n p x p π = sin + n dx p π p 0 −x − μk k=1 n π +∞ μk p dx = sin π p 0 xp + μk k=1 μk p 1 π +∞ sin dx. = p π p 0 x + μk μk =0
By using the software Mathematica, we have 1 x 1 xa 1 dx = ; 1 + ; − ), F (1, 2 1 xa + b b a a b 13
where 2 F1 (a0 , a1 ; b0 ; x) =
∞
k=0
(a0 )k (a1 )k xk k! (b0 )k
is a hypergeometric function with a > 1, b > 0
and (z)k = z(z + 1)(z + 2) · · · (z + k − 1) (see [1]). Again, using the software Mathematica, we get
1 xa 1 1 1 lim x 2 F1 (1, ; 1 + ; − ) = b1/a Γ(1 + )Γ(1 − ) x→+∞ a a b a a
and
1 xa 1 lim x 2 F1 (1, ; 1 + ; − ) = 0, x→0 a a b
where Γ(x) is the Gamma function. Since Γ(1 − x)Γ(x) =
π sin πx
for 0 < x < 1 and Γ(1 + x) = xΓ(x), we have 1 π +∞ pμk sin dx π p 0 xp + μk pμk π x 1 xa +∞ 1 = sin · ; 1 + ; − ) F (1, 2 1 π p μk p p μk 0 π 1/p 1 p 1 = sin · μk Γ(1 + )Γ(1 − ) π p p p π 1 1 1 1/p p sin · Γ( )Γ(1 − ) =μk π p p p p 1 π π 1/p 1/p sin · = μk . =μk π p sin πp Therefore, p +∞ xp φL (G, −xp ) π + n · sin dx p π 0 φL (G, −x ) p +∞ μk p 1 π sin dx = π p 0 xp + μk μk =0
=
μk =0
1/p
μk
=LELα (G). Let −1 < α = −1/p < 0 be an irrational number. Then 1 < p ∈ R\Q. For a graph G
14
−1/p with c components, LELα (G) = μk =0 μk . The integral in Theorem 4 (4) is p +∞ −x−p φL (G, −x−p ) π − c · sin dx −p π 0 φL (G, −x ) p +∞ n −p −x p π = sin − c dx π p 0 −x−p − μk k=1 n−c 1 π +∞ p dx = sin π p 0 1 + μ k xp k=1 n−c π +∞ μ−1 p k dx = sin p + μ−1 π p x 0 k k=1 n−c pμ−1 1 π +∞ k sin = −1 dx p π p 0 x + μ k k=1 =
=
=
n−c pμ−1 k
k=1 n−c k=1 n−c k=1
π
sin
π x 1 xa +∞ 1 · −1 2 F1 (1, ; 1 + ; − −1 ) p μk p p μk 0
p π −1/p 1 1 sin · μk Γ(1 + )Γ(1 − ) π p p p −1/p
μk
=LELα (G). Different from the case that α is an irrational number with 0 < |α| < 1, the integral formulas in Theorem 4 for the case |α| > 1 do not hold when α is irrational. Note that π p +∞ xp φL (G, −xp ) + n · sin dx p π 0 φL (G, −x ) p μk p 1 π +∞ sin dx. = π p 0 xp + μk μk =0
Then it follows from that the improper integral +∞ 1 dx, (0 < p < 1, a > 0) p x +a 0 diverges that the integral p π
+∞ p x φL (G, −xp ) φL (G, −xp ) 0
π + n · sin dx p
diverges. It can also be shown that the formulas in Theorem 5 (1) and (4), Theorem 6 (1) and (4) and Theorem 7 (1) and (4) hold when α is an irrational number with 0 < |α| < 1 and do not hold with |α| > 1. We omit the details here. Acknowledgements. The authors are grateful to Professor Ivan Gutman for his valuable comments on an early version of this paper. 15
References [1] G.E. Andrews, R. Askey, R. Roy, Special funtions, Cambridge University Press, Cambridge, 1999. [2] C.A. Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules, Proc. Cambridge Phil. Soc. 36 (1940) 201–203. [3] D.M. Cvetkovi´c, M. Doob, H. Sachs, Spectra of Graphs — Theory and Application, Academic Press, New York, 1980. [4] T.W. Gamelin, Complex Analysis, Springer-Verlag, New York, 2001. [5] I. Gutman, X. Li (Eds), Energies of Graphs — Theory and Applications, Math. Chem. Monogr., vol.17, ISBN 978-86-6009-033-3, 2016, p.III+290, Kragujevac. [6] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37. [7] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. [8] J. Liu, B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 (2008) 355–372. [9] L. Qiao, S. Zhang, B. Ning, J. Li, Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs I, J. Math. Anal. Appl. 435 (2016) 1249– 1261. [10] K. Yates, H¨ uckel Molecular Orbital Theory, Academic Press, New York, 1978. [11] B. Zhou, On sum of powers of the Laplacian eigenvalues of graphs, Linear Algebra Appl. 429 (2008) 2239–2246.
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