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Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs I
J. Math. Anal. Appl. 435 (2016) 1249–1261
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs I ✩ Lu Qiao, Shenggui Zhang ∗ , Bo Ning, Jing Li Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China
a r t i c l e
i n f o
Article history: Received 13 October 2014 Available online 23 October 2015 Submitted by P. Koskela Keywords: Laplacian matrix Coulson integral formula General Laplacian-energy-like invariant
a b s t r a c t n Let G be a simple graph. Its energy is defined as E(G) = k=1 |λk |, where λ1 , λ2 , . . . , λn are the eigenvalues of G. 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 by α 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 a Coulson-type integral formula for the general Laplacian-energy-like invariant for α = 1/p with p ∈ Z+ \{1}. This implies integral formulas for the Laplacian-energy-like invariant, the normalized incidence energy and the Laplacian incidence energy of graphs. © 2015 Elsevier Inc. All rights reserved.
1. Introduction All graphs considered in this paper are finite and simple. For terminology and notation not defined here, we refer the reader to Cvetković et al. [4]. Let G be a graph with n vertices and m edges. The eigenvalues of the adjacency matrix A(G) of G are said to be the eigenvalues of G and form the spectrum of G. We denote the eigenvalues of G by λ1 ≥ λ2 ≥ · · · ≥ λn in non-increasing order. The matrix L(G) = D(G) − A(G) is called the Laplacian matrix of G, where D(G) = diag(d1 , d2 , . . . , dn ) is the diagonal matrix of vertex degrees of G. It is well known that L(G) is a positive semi-definite symmetric matrix, and moreover 0 is the smallest eigenvalue of L(G). We denote the eigenvalues of L(G) by μ1 ≥ μ2 ≥ · · · ≥ μn = 0, which are called the Laplacian eigenvalues of G. ✩ Supported by NSFC (Nos. 11271300, 11201374 and 11571135), the Doctorate Foundation of Northwestern Polytechnical University (cx201326) and the Fundamental Research Funds for the Central Universities (3102014JCQ01073). * Corresponding author. E-mail addresses: [email protected] (L. Qiao), [email protected] (S. Zhang), [email protected] (B. Ning), [email protected] (J. Li).
http://dx.doi.org/10.1016/j.jmaa.2015.10.055 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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n The energy of a graph G is defined as E(G) = k=1 |λk |, which is derived from the total π-electron energy [13]. Graph energy has been studied extensively by many mathematicians and chemists, and there have been many results obtained on this invariant of graphs (see [8]). In the theory of graph energy there is an important result called the Coulson integral formula which makes it possible to calculate the energy of a graph without knowing its spectrum. For a graph G, its Coulson integral formula is 1 E(G) = π
+∞ ixφA (G, ix) n− dx, φA (G, ix)
−∞
where φA (G, x) is the characteristic polynomial of A(G) (called the characteristic polynomial of G). This formula was obtained by Coulson [2], and has many applications in the theory of graph energy (see [8]). For a graph G, since μk ≥ 0 for k = 1, 2, . . . , n, it would be trivial to define its Laplacian energy as n n k=1 |μk | = k=1 μk = 2m. Gutman and Zhou [6] defined the Laplacian energy of a graph G as n 2m LE(G) = μk − n . k=1
Later, Liu and Liu [9] introduced the Laplacian-energy-like invariant of G, which is similar to the definition of the graph energy, as
LEL(G) =
n √
μk .
k=1
This invariant has many similar properties as the energy of a graph. For more results on the Laplacianenergy-like invariant, we refer the reader to the references [7,9,12,16]. In [14], 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. Here we call this invariant the general Laplacianenergy-like invariant of graphs. 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 α μk =0 μk when μ1 = 0, and 0 when μ1 = 0. Obviously, LEL(G) = LEL 12 (G). In this paper, we obtain a Coulson-type integral formula for the general Laplacian-energy-like invariant of graphs in Section 3. Before that, in Section 2, we give some preliminaries. In Section 4, we present a Coulson-type integral formula for the general energy of polynomials, which is an extension of the general Laplacian-energy-like invariant of graphs, and show that it implies two known integral formulas for the normalized incidence energy and the Laplacian incidence energy. 2. Preliminaries We first introduce some basic concepts and results from complex analysis which will be used later. Let D be a bounded domain. The boundary of D is denoted by ∂D. The following two results in complex analysis are well known (see [5]).
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
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Lemma 1 (Cauchy’s theorem). Let D be a bounded domain with piecewise smooth boundary. If f (z) is analytic on D, and extends smoothly to ∂D, then f (z)dz = 0. ∂D
Lemma 2 (Cauchy integral formula). Let D be a bounded domain with piecewise smooth boundary. If f (z) is analytic on D, and extends smoothly to ∂D, then 1 f (z) = 2πi
∂D
f (ζ) dζ, z ∈ D. ζ −z
We also need the following simple lemmas. The proofs are omitted here. Lemma 3. Let Sr be the arc z(θ) = a0 + reiθ , θ1 ≤ θ ≤ θ2 , where r > 0 is a real number. 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 f (z)dz = i(θ2 − θ1 )λ.
lim
r→0+ Sr
Lemma 4. Suppose that Γ is a piecewise smooth curve. If f (z) is a continuous function on Γ, then f (z)dz ≤ |f (z)| · |dz|. Further, if Γ has length L, and |f (z)| ≤ M on Γ, then Γ Γ f (z)dz ≤ M L. Γ
Setting f (z) = 1 in Lemmas 1 and 2, we get 1 f (z0 ) = 2πi
∂D
dζ = ζ − z0
1, if z0 ∈ int(∂D) ; 0, if z0 ∈ ext(∂D),
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 computing, we get
k=0
k=1
zk zφ (z) z = =n+ . φ(z) z − zk z − zk n
n
k=1
k=1
That is zk zφ (z) −n= . φ(z) z − zk n
k=1
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If z1 , z2 , . . . , zn ∈ int(∂D), then we have 1 2πi
n n 1 zk zφ (z) − n dz = dz = zk . φ(z) 2πi z − zk ∂D k=1
∂D
k=1
3. Coulson-type integral formula for the general Laplacian-energy-like invariant of graphs Theorem 1. Let G be a graph of order n, φL (G, x) the characteristic polynomial of the Laplacian matrix L(G), and α = 1/p a number with p ∈ Z+ \{1}. Then the general Laplacian-energy-like invariant of G can be given by the following integral formula 1 LELα (G) = π
+∞ 0
π pxp φL (G, −xp ) + pn · sin dx. p φL (G, −x ) p
Proof. Let μ1 ≥ μ2 ≥ · · · ≥ μn = 0 be the roots of φL (G, x). It is well known that if G has c (< n) components, then the multiplicity of μn = 0 is c, which means that μ1 ≥ μ2 ≥ · · · ≥ μn−c > μn−c+1 = · · · = μn = 0. Let ϕL (G, z) = φL (G, z p ). Then we have ϕL (G, z) = (z ) · p c
n−c
(z − μk ) = z p
cp
·
k=1
n−c
p−1
k=1
t=0
1 p
(z − μk e
i 2tπ p
)
and zϕL (G, z) ϕL (G, z)
⎡ = z⎣
cp + z
= cp +
n−c
p−1
k=1
t=0
1 1
p−1 n−c
⎤ ⎦
z − μkp ei
2tπ p
z 1
2tπ
z − μkp ei p 1 2tπ p−1 n−c μkp ei p k=1 t=0
= cp +
1
z − μkp ei
k=1 t=0
= cp + (n − c)p +
2tπ p
+1 1
p−1 n−c
μkp ei
k=1 t=0
z − μkp ei
2tπ p
1
2tπ p
.
Therefore, 1
μ p ei p zϕL (G, z) k − pn = . 1 2tπ ϕL (G, z) p i t=0 z − μ e p n−c p−1
k=1
2tπ
k
Let Γ = ΓR ∪L1 ∪Sr ∪L2 be the positively (i.e., counterclockwise) oriented piecewise smooth Jordan curve 1
1
p (see Fig. 1), where R > max{μ1 , μ1p }, 0 < r < min{μn−c , μn−c }, ΓR is the contour {z(θ) = Reiθ , − πp ≤ π π iθ θ ≤ p }, L1 is the line {z(θ) = ρe , r ≤ ρ ≤ R, θ = p }, Sr is the curve {z(θ) = reiθ , − πp ≤ θ ≤ πp }, and 1
1
1
p L2 is the line {z(θ) = ρeiθ , r ≤ ρ ≤ R, θ = − πp }. Then the points μ1p , μ2p , . . . , μn−c are all in the interior 1
2tπ
1
2tπ
1
p of the contour Γ, and the points 0, μ1p ei p , μ2p ei p , . . . , μn−c ei of the contour Γ. It follows from Lemmas 1 and 2 that
2tπ p
, t = 1, 2, . . . , p − 1, are all in the exterior
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
Fig. 1. The contour Γ in Theorem 1.
1 2πi
1 2tπ n−c p−1 μkp ei p zϕL (G, z) 1 − pn dz = dz 1 2tπ ϕL (G, z) 2πi p i t=0 z − μ e p
Γ k=1
Γ
k
1 n−c μp 1 k = 1 dz 2πi z − μp
Γ k=1
=
n−c k=1
=
n−c
1 2πi
k
1 p
μk
1
Γ
z − μkp
dz
1 p
μk = LEL p1 (G).
k=1
Since the value of the integral 1 2πi
zϕL (G, z) − pn dz ϕL (G, z)
Γ
is independent of r and R, we obtain that 1 LELα (G) = 2πi =
1 2πi
lim
R→+∞ r→0+ Γ
lim
R→+∞ r→0+
zϕL (G, z) − pn dz ϕL (G, z)
zϕL (G, z) zϕL (G, z) − pn dz + − pn dz ϕL (G, z) ϕL (G, z) L1
ΓR
zϕL (G, z) zϕL (G, z) + − pn dz + − pn 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 1 2tπ p i p 1 − μk e z
1 1 2tπ 2tπ |ρ(cos θ + i sin θ) − μkp (cos 2tπ |z − μkp ei p | p + i sin p )| = = |z| ρ 2
ρ2 + μkp − 2 cos(θ − =
ρ
1
p 2tπ p )ρμk
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2
1
1
ρ2 + μkp − 2ρμkp |ρ − μkp | = . ρ ρ
≥ Thus,
1 n−c 2tπ p−1 zμkp ei p zϕL (G, z) z 1 ϕL (G, z) − pn ≤ 2tπ p i p k=1 t=0 z − μk e 1 2tπ p i p p−1 n−c μk e = 1 2tπ p i p k=1 t=0 − μk e 1 z ≤
p−1 n−c
1
ρμkp 1
k=1 t=0
|ρ − μkp |
.
Since p−1 n−c
1
ρμkp 1
k=1 t=0
|ρ − μkp |
→ 0, for ρ → 0,
by Lemma 3 we have zϕL (G, z) − pn dz → 0, for ρ → 0. ϕL (G, z)
Sr− 1
Suppose that ωkt = μkp ei
2tπ p
1
. Then |ωk0 | = |ωk1 | = · · · = |ωk(p−1) | = μkp . Thus,
1 p−1 2tπ μkp ei p z 1 2tπ t=0 z − μ p ei p k 1 1 1 p−1 p1 i 2tπ p−1 2tπ 2tπ 2tπ p i p p i p p−2 p i p p−2 p−1 μk e p (z + z μ e + · · · + z(μ e ) + (μ e ) ) k k k = z z p − μk t=0 p−1 ω (z p−1 + z p−2 ω + · · · + zω p−2 + ω p−1 ) kt kt kt kt = z p−μ z k t=0 p−1 p 2 p−1 ωkt kt ωkt + ωzkt + · · · + ωzp−2 + p−1 z . = μk 1 − zp t=0 Obviously, there exists Nk > 0, for k ∈ {1, 2, . . . , n − c}, such that |1 − μzpk | ≥ there exists Mk > 0, for k ∈ {1, 2, . . . , n − c}, such that
1 2
for |z| > Nk . For any ε > 0,
p−1 p−1 p ω 2 ω ω ε kt kt kt + · · · + p−2 + p−1 < 2n z z z t=0 p−1 r for |z| > Mk . Noting that t=0 ωkt = 0 unless r = p. Therefore, for any ε > 0, there exists N = max{N1 , N2 , . . . , Nn−c , M1 , M2 , . . . , Mn−c } such that
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
1 n−c 2tπ p−1 p i p zϕL (G, z) μk e z = − pn z 1 ϕL (G, z) 2tπ k=1 t=0 z − μ p ei p k p−1 2 n−c p−1 kt ωkt + ωzkt + · · · + ωzp−2 + = μk 1 − zp t=0
p ωkt z p−1
k=1
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p−1 p−1 n−c p−1 p n−c 2 ωkt ωkt ωkt ≤ 2 + · · · + p−2 + p−1 ωkt + 2 z z z t=0 t=0 k=1
<0+2
k=1
n−c k=1
ε <ε 2n
for |z| > N . By Lemma 4, it can be obtained that for any ε > 0 there exists N = max{N1 , N2 , . . . , Nn−c , M1 , M2 , . . . , Mn−c } such that zϕ (G, z) 2πR zϕL (G, z) − pn dz ≤ max L − pn ϕL (G, z) p z∈ΓR ϕL (G, z) ΓR
zϕL (G, z) 2π = max z − pn p z∈ΓR ϕL (G, z) <
2π ε p
for |z| > N . In other words, zϕL (G, z) − pn dz → 0, for |z| → +∞. ϕL (G, z) ΓR
Consequently, we have 1 LELα (G) = 2πi
lim
R→+∞ r→0+
zϕL (G, z) zϕL (G, z) − pn dz + − pn dz ϕL (G, z) ϕL (G, z) L1
ΓR
zϕL (G, z) zϕL (G, z) − pn dz + − pn dz + ϕL (G, z) ϕL (G, z) Sr−
=
=
1 2πi 1 2πi
L2
lim
R→+∞ r→0+
r
1 = 2πi
L1
r lim
R→+∞ r→0+
R +
zϕL (G, z) zϕL (G, z) − pn dz + − pn dz ϕL (G, z) ϕL (G, z)
R
L2 π π π p(ρei p )p φL (G, (ρei p )p )) − pn d(ρei p ) iπ p p φL (G, (ρe ) )
π π p(ρe−i p )p φL (G, (ρe−i p )p ) −i π p − pn d(ρe ) π φL (G, (ρe−i p )p ) r
lim
R→+∞ r→0+
R
−
π pρp φL (G, −ρp ) − pn ei p dρ φL (G, −ρp )
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R + r
1 = 2πi
R lim
R→+∞ r→0+
R − r
1 = 2πi 1 = π
r
π π pρp φL (G, −ρp ) + pn (cos + i sin )dρ p φL (G, −ρ ) p p
pρp φ (G, −ρp ) π π L + pn (cos(− ) + i sin(− ))dρ φL (G, −ρp ) p p R lim
R→+∞ r→0+ r
+∞ 0
π pρp φL (G, −ρp ) − pn e−i p dρ − φL (G, −ρp )
π pρp φL (G, −ρp ) + pn · 2i sin dρ p φL (G, −ρ ) p
π pxp φL (G, −xp ) + pn · sin dx. p φL (G, −x ) p
If G has n components, which means that μ1 = · · · = μn = 0, then φ(G, x) = xn . Thus, we have pxp n(−xp )n−1 pxp φL (G, −xp ) + pn = + pn = 0 p φL (G, −x ) (−xp )n and LEL p1 (G) =
n
1
k=1
μkp = 0. Therefore, 1 LELα (G) = π
+∞ 0
π pxp φL (G, −xp ) + pn · sin dx. φL (G, −xp ) p
This completes the proof. 2 Clearly, it is easy to obtain the following result from Theorem 1. Corollary 1. Let G be a graph of order n, and φL (G, x) the characteristic polynomial of the Laplacian matrix L(G). Then the Laplacian-energy-like invariant of G can be given by the following integral formula 1 LEL(G) = π
+∞ −∞
x2 φL (G, −x2 ) + n dx. φL (G, −x2 )
Proof. By Theorem 1, it can be obtained that 1 LEL(G) = LEL 12 (G) = π
+∞ 0
=
2 π
+∞
0
= which completes the proof. 2
1 π
π 2x2 φL (G, −x2 ) + 2n · sin dx φL (G, −x2 ) 2
x2 φ (G, −x2 ) L + n dx 2 φL (G, −x )
+∞
−∞
x2 φ (G, −x2 ) L + n dx, φL (G, −x2 )
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Let G1 and G2 be two graphs of same order. The Coulson–Jacobs formula (see [3,8]) gives the difference of their energies, that is 1 E(G1 ) − E(G2 ) = π
+∞ φ(G1 , ix) dx, ln φ(G2 , ix)
−∞
where φ(G, x) is the characteristic polynomial of the matrix A(G). Similar to this, we obtain the following theorem on the difference of the general Laplacian-energy-like invariant of two graphs. Theorem 2. Let G1 and G2 be two graphs of equal order. Then 1 LEL p1 (G1 ) − LEL p1 (G2 ) = π
+∞ φL (G1 , −xp ) · sin π dx, p ∈ Z+ \{1}, ln φL (G2 , −xp ) p 0
where φL (G, x) is the characteristic polynomial of the Laplacian matrix L(G). Proof. By Theorem 1, it can be obtained that LEL p1 (G1 ) − LEL p1 (G2 ) 1 = π
+∞ 0
=−
1 π
π pxp φL (G1 , −xp ) pxp φL (G2 , −xp ) − · sin dx φL (G1 , −xp ) φL (G2 , −xp ) p
+∞ φ (G , −xp ) φ (G , −xp ) π 1 2 − L x L · sin d(−xp ) φL (G1 , −xp ) φL (G2 , −xp ) p 0
=−
1 π
+∞
x sin 0
=−
sin πp π
φ (G , −xp ) π L 1 d ln p φL (G2 , −xp )
⎞ +∞ φ (G , −xp ) φ (G , −xp ) +∞ L 1 L 1 ⎝x ln − ln dx⎠ φL (G2 , −xp ) φL (G2 , −xp ) ⎛
0
0
and
x φL (G1 , −xp ) x→+∞ φL (G2 , −xp ) x φL (G1 , −xp ) − φL (G2 , −xp ) 1+ = lim x→+∞ φL (G2 , −xp ) φL (G2 ,−xp )[φL (G1 ,−xp )−φL (G2 ,−xp )]x φL (G1 , −xp ) − φL (G2 , −xp ) [φL (G1 ,−xp )−φL (G2 ,−xp )]φL (G2 ,−xp ) 1+ = lim x→+∞ φL (G2 , −xp ) lim
= e0 = 1, since the degree of [φL (G1 , −xp ) − φL (G2 , −xp )]x is less than the degree of φL (G2 , −xp ). Thus, φL (G1 , −xp ) = 0. lim x ln x→+∞ φL (G2 , −xp )
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Suppose that φL (Gj , x) = x
n−1
(x − μk (Gj )), j = 1, 2,
k=1
where μ1 (Gj ), . . . , μn (Gj ) are the Laplacian eigenvalues of Gj (j = 1, 2). Then n−1
−xp − μk (G1 ) φL (G1 , −xp ) = . φL (G2 , −xp ) −xp − μk (G2 ) k=1
Since lim x ln x = 0, we have x→0
n−1
−xp − μk (G1 ) φL (G1 , −xp ) = lim x ln = 0. lim x ln x→0 φL (G2 , −xp ) x→0 −xp − μk (G2 ) k=1
Therefore, 1 LEL p1 (G1 ) − LEL p1 (G2 ) = π
+∞ φL (G1 , −xp ) · sin π dx. ln φL (G2 , −xp ) p 0
This completes the proof. 2 Corollary 2. Let G be a simple graph of order n, and φL (G, x) = 1 LEL p1 (G) = 2π
n k=0
ak xn−k . Then
n 2 +∞ π −2 k pk x ln (−1) ak x · sin dx, p ∈ Z+ \{1}. p k=0
0
Proof. Noting that φ(Kn , x) = xn , by Theorem 2 we have +∞ n π ln ak (−xp )−k · sin dx p
1 LEL p1 (G) = π
k=0
0
=
0
1 π
+∞
1 = π
k=0
+∞ n π −2 k pk x ln (−1) ak x · sin dx p 0
1 = 2π
n π −p −k ln ak (−x ) · sin d(x−1 ) p
k=0
n 2 +∞ π x−2 ln (−1)k ak xpk · sin dx. p 0
k=0
Thus, the proof is complete. 2 Remark 1. According to Vieta’s formulas, we can obtain that (−1)k ak ≥ 0, for k = 1, 2, . . . , n. By the above result, we get that for two graphs G1 and G2 of equal order n, if (−1)k ak (G1 ) ≤ (−1)k ak (G2 ) for all 0 ≤ k ≤ n, then LEL p1 (G1 ) ≤ LEL p1 (G2 ). Zhou and Gutman [15] proved that if T is an n-vertex tree, then
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
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(−1)k ak (Sn ) ≤ (−1)k ak (T ) ≤ (−1)k ak (Pn ) for all 0 ≤ k ≤ n, where Sn and Pn are the star and path of n vertices respectively. This implies that LEL p1 (Sn ) ≤ LEL p1 (T ) ≤ LEL p1 (Pn ). 4. Integral formula for the general energy of polynomials We first extend the concept of the general Laplacian-energy-like invariant of graphs to complex polynomials. Definition 2. 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. By an analogous argument in the proof of Theorem 1, we can obtain the following result on the general energy of polynomials for α = 1/p with p ∈ Z+ \{1}. Theorem 3. Let φ(z) be a monic polynomial of degree n, whose roots are all non-negative real numbers, and α = 1/p a number with p ∈ Z+ \{1}. Then the general energy of φ(z) can be given by the following integral formula +∞
1 LELα (G) = π
π pxp φ (−xp ) + pn · sin dx. φ(−xp ) p
0
As an extension of the concept of graph energy, the energy E(M ) of a real n × m matrix M is defined by Nikiforov [10] as the sum of its singular values, which are the square roots of the eigenvalues of the square matrix M M T , where M T is the transpose of M . Let σ1 (M ), σn (M ), . . . , σn (M ) be the singular values of M . Then E(M ) =
n
σk (M ).
k=1
The normalized incidence energy NIE(G) of G, introduced by Cheng and Liu in [1], is the energy of the 1 1 ˆ matrix I(G) = D− 2 (G)I(G), where I(G) is the incidence matrix of G, D− 2 (G) is the diagonal matrix with √ 1 1 entries D− 2 (G)(k, k) = 1/ dk if dk = 0 and D− 2 (G)(k, k) = 0 otherwise. Then NIE(G) =
n k=1
ˆ σk (I(G)) =
n ˆ I(G) ˆ T ), λk (I(G) k=1
ˆ I(G) ˆ T ) (k = 1, . . . , n) are the eigenvalues of the matrix I(G) ˆ I(G) ˆ T . Obviously, I(G) ˆ I(G) ˆ T is where λk (I(G) a positive semi-definite matrix. Cheng and Liu [1] gave an integral formula for the normalized incidence energy of graphs. We find that their result is an immediate consequence of Theorem 3. Corollary 3. (See Cheng and Liu [1].) Let G be a graph of order n, and φ(x) the characteristic polynomial ˆ I(G) ˆ T . Then the normalized incidence energy of G can be given by the following integral of the matrix I(G) formula
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L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
1 NIE(G) = 2π
+∞
2n − ix
−∞
f (ix) dx, f (ix)
where f (x) = φ(x2 ). ˆ I(G) ˆ T is a positive semi-definite matrix, and all the roots of φ(x) are nonnegative. Note Proof. Clearly I(G) that f (ix) = 2 · (ix)φ ((ix)2 ) = 2ixφ (−x2 ) and 2n − ix
2ixφ (−x2 ) x2 φ (−x2 ) f (ix) = 2n − ix = 2[n + )]. f (ix) φ(−x2 ) φ(−x2 )
By Theorem 3, it can be obtained that
NIE(G) = E 12 (φ(x)) =
+∞
1 π
π 2x2 φ (−x2 ) + 2n · sin dx 2 φ(−x ) 2
0
=
+∞
x2 φ (−x2 )
2 π
φ(−x2 )
+ n dx
0
1 = π
+∞ −∞
1 = 2π 1 = 2π
x2 φ (−x2 ) + n dx φ(−x2 )
+∞ x2 φ (−x2 ) + n dx 2 φ(−x2 )
−∞
+∞ −∞
2n − ix
f (ix) dx. f (ix)
Thus, the proof is complete. 2 By assigning an arbitrary orientation to the edges of G with vertex set V (G) = {v1 , . . . , vn }, the vertex-arc → − → − incidence matrix S(G) = (sie ) of G is defined as ⎧ ⎪ if vi is the head of e; ⎨ 1, sie = −1, if vi is the tail of e; ⎪ ⎩ 0, otherwise. → −
→ −
→ −
→ −
The normalized oriented incidence matrix of G, denoted by S (G), is defined as S (G) = D− 2 (G)S(G). The normalized Laplacian matrix of G, denoted by NL(G) = (lij ), is the matrix with entries ⎧ ⎪ 1, if i = j and di = 0; ⎪ ⎨ 1 $ lij = − di dj , if vi and vj are adjacent in G; ⎪ ⎪ ⎩ 0, otherwise.
1
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261 → −
→ −
1261
→ −
Clearly, NL(G) = S (G)S (G)T , where G is an arbitrary oriented graph of G. The Laplacian incidence energy LIE(G) of G, introduced by Shi and Wang in [11], is defined as LIE(G) =
n
→ −
σk (S (G)) =
k=1
n $ λk (NL(G)), k=1
where λk (NL(G)) (k = 1, . . . , n) are the eigenvalues of NL(G). Shi and Wang [11] gave an integral formula for Laplacian incidence energy of graphs. Their result is also an immediate consequence of Theorem 3. Corollary 4. (See Shi and Wang [11].) Let G be a graph of order n, and φ(x) the characteristic polynomial of the normalized Laplacian matrix NL(G) of G. Then the Laplacian incidence energy of G can be given by the following integral formula 1 LIE(G) = π
+∞
n+ −∞
x2 φ (−x2 ) dx. φ(−x2 )
We omit the proof of this corollary here. References [1] B. Cheng, B. Liu, The normalized incidence energy of a graph, Linear Algebra Appl. 438 (2013) 4510–4519. [2] C.A. Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules, Proc. Camb. Philol. Soc. 36 (1940) 201–203. [3] C.A. Coulson, J. Jacobs, Conjugation across a single bond, J. Chem. Soc. (1949) 2805–2812. [4] D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs — Theory and Application, Academic Press, New York, 1980. [5] T.W. Gamelin, Complex Analysis, Springer-Verlag, New York, 2001. [6] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37. [7] I. Gutman, B. Zhou, B. Furtula, The Laplacian-energy like invariant is an energy like invariant, MATCH Commun. Math. Comput. Chem. 64 (2010) 85–96. [8] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. [9] J. Liu, B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 (2008) 355–372. [10] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 326 (2007) 1472–1475. [11] L. Shi, H. Wang, The Laplacian incidence energy of graphs, Linear Algebra Appl. 439 (2013) 4056–4062. [12] D. Stevanović, Laplacian-like energy of trees, MATCH Commun. Math. Comput. Chem. 61 (2009) 407–417. [13] K. Yates, Hückel Molecular Orbital Theory, Academic Press, New York, 1978. [14] B. Zhou, On sum of powers of the Laplacian eigenvalues of graphs, Linear Algebra Appl. 429 (2008) 2239–2246. [15] B. Zhou, I. Gutman, A connection between ordinary and Laplacian spectra of bipartite graphs, Linear Multilinear Algebra 56 (2008) 305–310. [16] B. Zhu, The Laplacian-energy like of graphs, Appl. Math. Lett. 24 (2011) 1604–1607.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs I ✩ Lu Qiao, Shenggui Zhang ∗ , Bo Ning, Jing Li Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China
a r t i c l e
i n f o
Article history: Received 13 October 2014 Available online 23 October 2015 Submitted by P. Koskela Keywords: Laplacian matrix Coulson integral formula General Laplacian-energy-like invariant
a b s t r a c t n Let G be a simple graph. Its energy is defined as E(G) = k=1 |λk |, where λ1 , λ2 , . . . , λn are the eigenvalues of G. 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 by α 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 a Coulson-type integral formula for the general Laplacian-energy-like invariant for α = 1/p with p ∈ Z+ \{1}. This implies integral formulas for the Laplacian-energy-like invariant, the normalized incidence energy and the Laplacian incidence energy of graphs. © 2015 Elsevier Inc. All rights reserved.
1. Introduction All graphs considered in this paper are finite and simple. For terminology and notation not defined here, we refer the reader to Cvetković et al. [4]. Let G be a graph with n vertices and m edges. The eigenvalues of the adjacency matrix A(G) of G are said to be the eigenvalues of G and form the spectrum of G. We denote the eigenvalues of G by λ1 ≥ λ2 ≥ · · · ≥ λn in non-increasing order. The matrix L(G) = D(G) − A(G) is called the Laplacian matrix of G, where D(G) = diag(d1 , d2 , . . . , dn ) is the diagonal matrix of vertex degrees of G. It is well known that L(G) is a positive semi-definite symmetric matrix, and moreover 0 is the smallest eigenvalue of L(G). We denote the eigenvalues of L(G) by μ1 ≥ μ2 ≥ · · · ≥ μn = 0, which are called the Laplacian eigenvalues of G. ✩ Supported by NSFC (Nos. 11271300, 11201374 and 11571135), the Doctorate Foundation of Northwestern Polytechnical University (cx201326) and the Fundamental Research Funds for the Central Universities (3102014JCQ01073). * Corresponding author. E-mail addresses: [email protected] (L. Qiao), [email protected] (S. Zhang), [email protected] (B. Ning), [email protected] (J. Li).
http://dx.doi.org/10.1016/j.jmaa.2015.10.055 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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n The energy of a graph G is defined as E(G) = k=1 |λk |, which is derived from the total π-electron energy [13]. Graph energy has been studied extensively by many mathematicians and chemists, and there have been many results obtained on this invariant of graphs (see [8]). In the theory of graph energy there is an important result called the Coulson integral formula which makes it possible to calculate the energy of a graph without knowing its spectrum. For a graph G, its Coulson integral formula is 1 E(G) = π
+∞ ixφA (G, ix) n− dx, φA (G, ix)
−∞
where φA (G, x) is the characteristic polynomial of A(G) (called the characteristic polynomial of G). This formula was obtained by Coulson [2], and has many applications in the theory of graph energy (see [8]). For a graph G, since μk ≥ 0 for k = 1, 2, . . . , n, it would be trivial to define its Laplacian energy as n n k=1 |μk | = k=1 μk = 2m. Gutman and Zhou [6] defined the Laplacian energy of a graph G as n 2m LE(G) = μk − n . k=1
Later, Liu and Liu [9] introduced the Laplacian-energy-like invariant of G, which is similar to the definition of the graph energy, as
LEL(G) =
n √
μk .
k=1
This invariant has many similar properties as the energy of a graph. For more results on the Laplacianenergy-like invariant, we refer the reader to the references [7,9,12,16]. In [14], 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. Here we call this invariant the general Laplacianenergy-like invariant of graphs. 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 α μk =0 μk when μ1 = 0, and 0 when μ1 = 0. Obviously, LEL(G) = LEL 12 (G). In this paper, we obtain a Coulson-type integral formula for the general Laplacian-energy-like invariant of graphs in Section 3. Before that, in Section 2, we give some preliminaries. In Section 4, we present a Coulson-type integral formula for the general energy of polynomials, which is an extension of the general Laplacian-energy-like invariant of graphs, and show that it implies two known integral formulas for the normalized incidence energy and the Laplacian incidence energy. 2. Preliminaries We first introduce some basic concepts and results from complex analysis which will be used later. Let D be a bounded domain. The boundary of D is denoted by ∂D. The following two results in complex analysis are well known (see [5]).
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
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Lemma 1 (Cauchy’s theorem). Let D be a bounded domain with piecewise smooth boundary. If f (z) is analytic on D, and extends smoothly to ∂D, then f (z)dz = 0. ∂D
Lemma 2 (Cauchy integral formula). Let D be a bounded domain with piecewise smooth boundary. If f (z) is analytic on D, and extends smoothly to ∂D, then 1 f (z) = 2πi
∂D
f (ζ) dζ, z ∈ D. ζ −z
We also need the following simple lemmas. The proofs are omitted here. Lemma 3. Let Sr be the arc z(θ) = a0 + reiθ , θ1 ≤ θ ≤ θ2 , where r > 0 is a real number. 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 f (z)dz = i(θ2 − θ1 )λ.
lim
r→0+ Sr
Lemma 4. Suppose that Γ is a piecewise smooth curve. If f (z) is a continuous function on Γ, then f (z)dz ≤ |f (z)| · |dz|. Further, if Γ has length L, and |f (z)| ≤ M on Γ, then Γ Γ f (z)dz ≤ M L. Γ
Setting f (z) = 1 in Lemmas 1 and 2, we get 1 f (z0 ) = 2πi
∂D
dζ = ζ − z0
1, if z0 ∈ int(∂D) ; 0, if z0 ∈ ext(∂D),
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 computing, we get
k=0
k=1
zk zφ (z) z = =n+ . φ(z) z − zk z − zk n
n
k=1
k=1
That is zk zφ (z) −n= . φ(z) z − zk n
k=1
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L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
If z1 , z2 , . . . , zn ∈ int(∂D), then we have 1 2πi
n n 1 zk zφ (z) − n dz = dz = zk . φ(z) 2πi z − zk ∂D k=1
∂D
k=1
3. Coulson-type integral formula for the general Laplacian-energy-like invariant of graphs Theorem 1. Let G be a graph of order n, φL (G, x) the characteristic polynomial of the Laplacian matrix L(G), and α = 1/p a number with p ∈ Z+ \{1}. Then the general Laplacian-energy-like invariant of G can be given by the following integral formula 1 LELα (G) = π
+∞ 0
π pxp φL (G, −xp ) + pn · sin dx. p φL (G, −x ) p
Proof. Let μ1 ≥ μ2 ≥ · · · ≥ μn = 0 be the roots of φL (G, x). It is well known that if G has c (< n) components, then the multiplicity of μn = 0 is c, which means that μ1 ≥ μ2 ≥ · · · ≥ μn−c > μn−c+1 = · · · = μn = 0. Let ϕL (G, z) = φL (G, z p ). Then we have ϕL (G, z) = (z ) · p c
n−c
(z − μk ) = z p
cp
·
k=1
n−c
p−1
k=1
t=0
1 p
(z − μk e
i 2tπ p
)
and zϕL (G, z) ϕL (G, z)
⎡ = z⎣
cp + z
= cp +
n−c
p−1
k=1
t=0
1 1
p−1 n−c
⎤ ⎦
z − μkp ei
2tπ p
z 1
2tπ
z − μkp ei p 1 2tπ p−1 n−c μkp ei p k=1 t=0
= cp +
1
z − μkp ei
k=1 t=0
= cp + (n − c)p +
2tπ p
+1 1
p−1 n−c
μkp ei
k=1 t=0
z − μkp ei
2tπ p
1
2tπ p
.
Therefore, 1
μ p ei p zϕL (G, z) k − pn = . 1 2tπ ϕL (G, z) p i t=0 z − μ e p n−c p−1
k=1
2tπ
k
Let Γ = ΓR ∪L1 ∪Sr ∪L2 be the positively (i.e., counterclockwise) oriented piecewise smooth Jordan curve 1
1
p (see Fig. 1), where R > max{μ1 , μ1p }, 0 < r < min{μn−c , μn−c }, ΓR is the contour {z(θ) = Reiθ , − πp ≤ π π iθ θ ≤ p }, L1 is the line {z(θ) = ρe , r ≤ ρ ≤ R, θ = p }, Sr is the curve {z(θ) = reiθ , − πp ≤ θ ≤ πp }, and 1
1
1
p L2 is the line {z(θ) = ρeiθ , r ≤ ρ ≤ R, θ = − πp }. Then the points μ1p , μ2p , . . . , μn−c are all in the interior 1
2tπ
1
2tπ
1
p of the contour Γ, and the points 0, μ1p ei p , μ2p ei p , . . . , μn−c ei of the contour Γ. It follows from Lemmas 1 and 2 that
2tπ p
, t = 1, 2, . . . , p − 1, are all in the exterior
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
Fig. 1. The contour Γ in Theorem 1.
1 2πi
1 2tπ n−c p−1 μkp ei p zϕL (G, z) 1 − pn dz = dz 1 2tπ ϕL (G, z) 2πi p i t=0 z − μ e p
Γ k=1
Γ
k
1 n−c μp 1 k = 1 dz 2πi z − μp
Γ k=1
=
n−c k=1
=
n−c
1 2πi
k
1 p
μk
1
Γ
z − μkp
dz
1 p
μk = LEL p1 (G).
k=1
Since the value of the integral 1 2πi
zϕL (G, z) − pn dz ϕL (G, z)
Γ
is independent of r and R, we obtain that 1 LELα (G) = 2πi =
1 2πi
lim
R→+∞ r→0+ Γ
lim
R→+∞ r→0+
zϕL (G, z) − pn dz ϕL (G, z)
zϕL (G, z) zϕL (G, z) − pn dz + − pn dz ϕL (G, z) ϕL (G, z) L1
ΓR
zϕL (G, z) zϕL (G, z) + − pn dz + − pn 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 1 2tπ p i p 1 − μk e z
1 1 2tπ 2tπ |ρ(cos θ + i sin θ) − μkp (cos 2tπ |z − μkp ei p | p + i sin p )| = = |z| ρ 2
ρ2 + μkp − 2 cos(θ − =
ρ
1
p 2tπ p )ρμk
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L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
1254
2
1
1
ρ2 + μkp − 2ρμkp |ρ − μkp | = . ρ ρ
≥ Thus,
1 n−c 2tπ p−1 zμkp ei p zϕL (G, z) z 1 ϕL (G, z) − pn ≤ 2tπ p i p k=1 t=0 z − μk e 1 2tπ p i p p−1 n−c μk e = 1 2tπ p i p k=1 t=0 − μk e 1 z ≤
p−1 n−c
1
ρμkp 1
k=1 t=0
|ρ − μkp |
.
Since p−1 n−c
1
ρμkp 1
k=1 t=0
|ρ − μkp |
→ 0, for ρ → 0,
by Lemma 3 we have zϕL (G, z) − pn dz → 0, for ρ → 0. ϕL (G, z)
Sr− 1
Suppose that ωkt = μkp ei
2tπ p
1
. Then |ωk0 | = |ωk1 | = · · · = |ωk(p−1) | = μkp . Thus,
1 p−1 2tπ μkp ei p z 1 2tπ t=0 z − μ p ei p k 1 1 1 p−1 p1 i 2tπ p−1 2tπ 2tπ 2tπ p i p p i p p−2 p i p p−2 p−1 μk e p (z + z μ e + · · · + z(μ e ) + (μ e ) ) k k k = z z p − μk t=0 p−1 ω (z p−1 + z p−2 ω + · · · + zω p−2 + ω p−1 ) kt kt kt kt = z p−μ z k t=0 p−1 p 2 p−1 ωkt kt ωkt + ωzkt + · · · + ωzp−2 + p−1 z . = μk 1 − zp t=0 Obviously, there exists Nk > 0, for k ∈ {1, 2, . . . , n − c}, such that |1 − μzpk | ≥ there exists Mk > 0, for k ∈ {1, 2, . . . , n − c}, such that
1 2
for |z| > Nk . For any ε > 0,
p−1 p−1 p ω 2 ω ω ε kt kt kt + · · · + p−2 + p−1 < 2n z z z t=0 p−1 r for |z| > Mk . Noting that t=0 ωkt = 0 unless r = p. Therefore, for any ε > 0, there exists N = max{N1 , N2 , . . . , Nn−c , M1 , M2 , . . . , Mn−c } such that
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
1 n−c 2tπ p−1 p i p zϕL (G, z) μk e z = − pn z 1 ϕL (G, z) 2tπ k=1 t=0 z − μ p ei p k p−1 2 n−c p−1 kt ωkt + ωzkt + · · · + ωzp−2 + = μk 1 − zp t=0
p ωkt z p−1
k=1
1255
p−1 p−1 n−c p−1 p n−c 2 ωkt ωkt ωkt ≤ 2 + · · · + p−2 + p−1 ωkt + 2 z z z t=0 t=0 k=1
<0+2
k=1
n−c k=1
ε <ε 2n
for |z| > N . By Lemma 4, it can be obtained that for any ε > 0 there exists N = max{N1 , N2 , . . . , Nn−c , M1 , M2 , . . . , Mn−c } such that zϕ (G, z) 2πR zϕL (G, z) − pn dz ≤ max L − pn ϕL (G, z) p z∈ΓR ϕL (G, z) ΓR
zϕL (G, z) 2π = max z − pn p z∈ΓR ϕL (G, z) <
2π ε p
for |z| > N . In other words, zϕL (G, z) − pn dz → 0, for |z| → +∞. ϕL (G, z) ΓR
Consequently, we have 1 LELα (G) = 2πi
lim
R→+∞ r→0+
zϕL (G, z) zϕL (G, z) − pn dz + − pn dz ϕL (G, z) ϕL (G, z) L1
ΓR
zϕL (G, z) zϕL (G, z) − pn dz + − pn dz + ϕL (G, z) ϕL (G, z) Sr−
=
=
1 2πi 1 2πi
L2
lim
R→+∞ r→0+
r
1 = 2πi
L1
r lim
R→+∞ r→0+
R +
zϕL (G, z) zϕL (G, z) − pn dz + − pn dz ϕL (G, z) ϕL (G, z)
R
L2 π π π p(ρei p )p φL (G, (ρei p )p )) − pn d(ρei p ) iπ p p φL (G, (ρe ) )
π π p(ρe−i p )p φL (G, (ρe−i p )p ) −i π p − pn d(ρe ) π φL (G, (ρe−i p )p ) r
lim
R→+∞ r→0+
R
−
π pρp φL (G, −ρp ) − pn ei p dρ φL (G, −ρp )
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
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R + r
1 = 2πi
R lim
R→+∞ r→0+
R − r
1 = 2πi 1 = π
r
π π pρp φL (G, −ρp ) + pn (cos + i sin )dρ p φL (G, −ρ ) p p
pρp φ (G, −ρp ) π π L + pn (cos(− ) + i sin(− ))dρ φL (G, −ρp ) p p R lim
R→+∞ r→0+ r
+∞ 0
π pρp φL (G, −ρp ) − pn e−i p dρ − φL (G, −ρp )
π pρp φL (G, −ρp ) + pn · 2i sin dρ p φL (G, −ρ ) p
π pxp φL (G, −xp ) + pn · sin dx. p φL (G, −x ) p
If G has n components, which means that μ1 = · · · = μn = 0, then φ(G, x) = xn . Thus, we have pxp n(−xp )n−1 pxp φL (G, −xp ) + pn = + pn = 0 p φL (G, −x ) (−xp )n and LEL p1 (G) =
n
1
k=1
μkp = 0. Therefore, 1 LELα (G) = π
+∞ 0
π pxp φL (G, −xp ) + pn · sin dx. φL (G, −xp ) p
This completes the proof. 2 Clearly, it is easy to obtain the following result from Theorem 1. Corollary 1. Let G be a graph of order n, and φL (G, x) the characteristic polynomial of the Laplacian matrix L(G). Then the Laplacian-energy-like invariant of G can be given by the following integral formula 1 LEL(G) = π
+∞ −∞
x2 φL (G, −x2 ) + n dx. φL (G, −x2 )
Proof. By Theorem 1, it can be obtained that 1 LEL(G) = LEL 12 (G) = π
+∞ 0
=
2 π
+∞
0
= which completes the proof. 2
1 π
π 2x2 φL (G, −x2 ) + 2n · sin dx φL (G, −x2 ) 2
x2 φ (G, −x2 ) L + n dx 2 φL (G, −x )
+∞
−∞
x2 φ (G, −x2 ) L + n dx, φL (G, −x2 )
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
1257
Let G1 and G2 be two graphs of same order. The Coulson–Jacobs formula (see [3,8]) gives the difference of their energies, that is 1 E(G1 ) − E(G2 ) = π
+∞ φ(G1 , ix) dx, ln φ(G2 , ix)
−∞
where φ(G, x) is the characteristic polynomial of the matrix A(G). Similar to this, we obtain the following theorem on the difference of the general Laplacian-energy-like invariant of two graphs. Theorem 2. Let G1 and G2 be two graphs of equal order. Then 1 LEL p1 (G1 ) − LEL p1 (G2 ) = π
+∞ φL (G1 , −xp ) · sin π dx, p ∈ Z+ \{1}, ln φL (G2 , −xp ) p 0
where φL (G, x) is the characteristic polynomial of the Laplacian matrix L(G). Proof. By Theorem 1, it can be obtained that LEL p1 (G1 ) − LEL p1 (G2 ) 1 = π
+∞ 0
=−
1 π
π pxp φL (G1 , −xp ) pxp φL (G2 , −xp ) − · sin dx φL (G1 , −xp ) φL (G2 , −xp ) p
+∞ φ (G , −xp ) φ (G , −xp ) π 1 2 − L x L · sin d(−xp ) φL (G1 , −xp ) φL (G2 , −xp ) p 0
=−
1 π
+∞
x sin 0
=−
sin πp π
φ (G , −xp ) π L 1 d ln p φL (G2 , −xp )
⎞ +∞ φ (G , −xp ) φ (G , −xp ) +∞ L 1 L 1 ⎝x ln − ln dx⎠ φL (G2 , −xp ) φL (G2 , −xp ) ⎛
0
0
and
x φL (G1 , −xp ) x→+∞ φL (G2 , −xp ) x φL (G1 , −xp ) − φL (G2 , −xp ) 1+ = lim x→+∞ φL (G2 , −xp ) φL (G2 ,−xp )[φL (G1 ,−xp )−φL (G2 ,−xp )]x φL (G1 , −xp ) − φL (G2 , −xp ) [φL (G1 ,−xp )−φL (G2 ,−xp )]φL (G2 ,−xp ) 1+ = lim x→+∞ φL (G2 , −xp ) lim
= e0 = 1, since the degree of [φL (G1 , −xp ) − φL (G2 , −xp )]x is less than the degree of φL (G2 , −xp ). Thus, φL (G1 , −xp ) = 0. lim x ln x→+∞ φL (G2 , −xp )
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
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Suppose that φL (Gj , x) = x
n−1
(x − μk (Gj )), j = 1, 2,
k=1
where μ1 (Gj ), . . . , μn (Gj ) are the Laplacian eigenvalues of Gj (j = 1, 2). Then n−1
−xp − μk (G1 ) φL (G1 , −xp ) = . φL (G2 , −xp ) −xp − μk (G2 ) k=1
Since lim x ln x = 0, we have x→0
n−1
−xp − μk (G1 ) φL (G1 , −xp ) = lim x ln = 0. lim x ln x→0 φL (G2 , −xp ) x→0 −xp − μk (G2 ) k=1
Therefore, 1 LEL p1 (G1 ) − LEL p1 (G2 ) = π
+∞ φL (G1 , −xp ) · sin π dx. ln φL (G2 , −xp ) p 0
This completes the proof. 2 Corollary 2. Let G be a simple graph of order n, and φL (G, x) = 1 LEL p1 (G) = 2π
n k=0
ak xn−k . Then
n 2 +∞ π −2 k pk x ln (−1) ak x · sin dx, p ∈ Z+ \{1}. p k=0
0
Proof. Noting that φ(Kn , x) = xn , by Theorem 2 we have +∞ n π ln ak (−xp )−k · sin dx p
1 LEL p1 (G) = π
k=0
0
=
0
1 π
+∞
1 = π
k=0
+∞ n π −2 k pk x ln (−1) ak x · sin dx p 0
1 = 2π
n π −p −k ln ak (−x ) · sin d(x−1 ) p
k=0
n 2 +∞ π x−2 ln (−1)k ak xpk · sin dx. p 0
k=0
Thus, the proof is complete. 2 Remark 1. According to Vieta’s formulas, we can obtain that (−1)k ak ≥ 0, for k = 1, 2, . . . , n. By the above result, we get that for two graphs G1 and G2 of equal order n, if (−1)k ak (G1 ) ≤ (−1)k ak (G2 ) for all 0 ≤ k ≤ n, then LEL p1 (G1 ) ≤ LEL p1 (G2 ). Zhou and Gutman [15] proved that if T is an n-vertex tree, then
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
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(−1)k ak (Sn ) ≤ (−1)k ak (T ) ≤ (−1)k ak (Pn ) for all 0 ≤ k ≤ n, where Sn and Pn are the star and path of n vertices respectively. This implies that LEL p1 (Sn ) ≤ LEL p1 (T ) ≤ LEL p1 (Pn ). 4. Integral formula for the general energy of polynomials We first extend the concept of the general Laplacian-energy-like invariant of graphs to complex polynomials. Definition 2. 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. By an analogous argument in the proof of Theorem 1, we can obtain the following result on the general energy of polynomials for α = 1/p with p ∈ Z+ \{1}. Theorem 3. Let φ(z) be a monic polynomial of degree n, whose roots are all non-negative real numbers, and α = 1/p a number with p ∈ Z+ \{1}. Then the general energy of φ(z) can be given by the following integral formula +∞
1 LELα (G) = π
π pxp φ (−xp ) + pn · sin dx. φ(−xp ) p
0
As an extension of the concept of graph energy, the energy E(M ) of a real n × m matrix M is defined by Nikiforov [10] as the sum of its singular values, which are the square roots of the eigenvalues of the square matrix M M T , where M T is the transpose of M . Let σ1 (M ), σn (M ), . . . , σn (M ) be the singular values of M . Then E(M ) =
n
σk (M ).
k=1
The normalized incidence energy NIE(G) of G, introduced by Cheng and Liu in [1], is the energy of the 1 1 ˆ matrix I(G) = D− 2 (G)I(G), where I(G) is the incidence matrix of G, D− 2 (G) is the diagonal matrix with √ 1 1 entries D− 2 (G)(k, k) = 1/ dk if dk = 0 and D− 2 (G)(k, k) = 0 otherwise. Then NIE(G) =
n k=1
ˆ σk (I(G)) =
n ˆ I(G) ˆ T ), λk (I(G) k=1
ˆ I(G) ˆ T ) (k = 1, . . . , n) are the eigenvalues of the matrix I(G) ˆ I(G) ˆ T . Obviously, I(G) ˆ I(G) ˆ T is where λk (I(G) a positive semi-definite matrix. Cheng and Liu [1] gave an integral formula for the normalized incidence energy of graphs. We find that their result is an immediate consequence of Theorem 3. Corollary 3. (See Cheng and Liu [1].) Let G be a graph of order n, and φ(x) the characteristic polynomial ˆ I(G) ˆ T . Then the normalized incidence energy of G can be given by the following integral of the matrix I(G) formula
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L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261
1 NIE(G) = 2π
+∞
2n − ix
−∞
f (ix) dx, f (ix)
where f (x) = φ(x2 ). ˆ I(G) ˆ T is a positive semi-definite matrix, and all the roots of φ(x) are nonnegative. Note Proof. Clearly I(G) that f (ix) = 2 · (ix)φ ((ix)2 ) = 2ixφ (−x2 ) and 2n − ix
2ixφ (−x2 ) x2 φ (−x2 ) f (ix) = 2n − ix = 2[n + )]. f (ix) φ(−x2 ) φ(−x2 )
By Theorem 3, it can be obtained that
NIE(G) = E 12 (φ(x)) =
+∞
1 π
π 2x2 φ (−x2 ) + 2n · sin dx 2 φ(−x ) 2
0
=
+∞
x2 φ (−x2 )
2 π
φ(−x2 )
+ n dx
0
1 = π
+∞ −∞
1 = 2π 1 = 2π
x2 φ (−x2 ) + n dx φ(−x2 )
+∞ x2 φ (−x2 ) + n dx 2 φ(−x2 )
−∞
+∞ −∞
2n − ix
f (ix) dx. f (ix)
Thus, the proof is complete. 2 By assigning an arbitrary orientation to the edges of G with vertex set V (G) = {v1 , . . . , vn }, the vertex-arc → − → − incidence matrix S(G) = (sie ) of G is defined as ⎧ ⎪ if vi is the head of e; ⎨ 1, sie = −1, if vi is the tail of e; ⎪ ⎩ 0, otherwise. → −
→ −
→ −
→ −
The normalized oriented incidence matrix of G, denoted by S (G), is defined as S (G) = D− 2 (G)S(G). The normalized Laplacian matrix of G, denoted by NL(G) = (lij ), is the matrix with entries ⎧ ⎪ 1, if i = j and di = 0; ⎪ ⎨ 1 $ lij = − di dj , if vi and vj are adjacent in G; ⎪ ⎪ ⎩ 0, otherwise.
1
L. Qiao et al. / J. Math. Anal. Appl. 435 (2016) 1249–1261 → −
→ −
1261
→ −
Clearly, NL(G) = S (G)S (G)T , where G is an arbitrary oriented graph of G. The Laplacian incidence energy LIE(G) of G, introduced by Shi and Wang in [11], is defined as LIE(G) =
n
→ −
σk (S (G)) =
k=1
n $ λk (NL(G)), k=1
where λk (NL(G)) (k = 1, . . . , n) are the eigenvalues of NL(G). Shi and Wang [11] gave an integral formula for Laplacian incidence energy of graphs. Their result is also an immediate consequence of Theorem 3. Corollary 4. (See Shi and Wang [11].) Let G be a graph of order n, and φ(x) the characteristic polynomial of the normalized Laplacian matrix NL(G) of G. Then the Laplacian incidence energy of G can be given by the following integral formula 1 LIE(G) = π
+∞
n+ −∞
x2 φ (−x2 ) dx. φ(−x2 )
We omit the proof of this corollary here. References [1] B. Cheng, B. Liu, The normalized incidence energy of a graph, Linear Algebra Appl. 438 (2013) 4510–4519. [2] C.A. Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules, Proc. Camb. Philol. Soc. 36 (1940) 201–203. [3] C.A. Coulson, J. Jacobs, Conjugation across a single bond, J. Chem. Soc. (1949) 2805–2812. [4] D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs — Theory and Application, Academic Press, New York, 1980. [5] T.W. Gamelin, Complex Analysis, Springer-Verlag, New York, 2001. [6] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37. [7] I. Gutman, B. Zhou, B. Furtula, The Laplacian-energy like invariant is an energy like invariant, MATCH Commun. Math. Comput. Chem. 64 (2010) 85–96. [8] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. [9] J. Liu, B. Liu, A Laplacian-energy-like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 (2008) 355–372. [10] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 326 (2007) 1472–1475. [11] L. Shi, H. Wang, The Laplacian incidence energy of graphs, Linear Algebra Appl. 439 (2013) 4056–4062. [12] D. Stevanović, Laplacian-like energy of trees, MATCH Commun. Math. Comput. Chem. 61 (2009) 407–417. [13] K. Yates, Hückel Molecular Orbital Theory, Academic Press, New York, 1978. [14] B. Zhou, On sum of powers of the Laplacian eigenvalues of graphs, Linear Algebra Appl. 429 (2008) 2239–2246. [15] B. Zhou, I. Gutman, A connection between ordinary and Laplacian spectra of bipartite graphs, Linear Multilinear Algebra 56 (2008) 305–310. [16] B. Zhu, The Laplacian-energy like of graphs, Appl. Math. Lett. 24 (2011) 1604–1607.