Vol. 23, No. 2 (2017), pp. 21–31.
BOUNDS ON ENERGY AND LAPLACIAN ENERGY OF
GRAPHS
Sridhara G
1,2, M.R.Rajesh kanna
3,41,3Post Graduate Department of Mathematics,
Maharani’s Science College for Women, J. L. B. Road, Mysore - 570 005, India.
2
Research Scholar, Research and Development Centre, Bharathiar University, Coimbatore 641 046, India
srsrig@gmail.com
4
Government First Grade College, Bettampady, Puttur,D.K, India
mr.rajeshkanna@gmail.com
Abstract. LetGbe simple graph withnvertices andmedges. The energyE(G) of G, denoted by E(G), is defined to be the sum of the absolute values of the eigenvalues ofG. In this paper, we present two new upper bounds for energy of a graph, one in terms ofm,nand another in terms of largest absolute eigenvalue and the smallest absolute eigenvalue. The paper also contains upper bounds for Laplacian energy of graph.
Key words and Phrases: Adjacency matrix, Laplacian matrix, Energy of graph, Laplacian energy of graph.
Abstrak. Misalkan G adalah graf sederhana denganntitik danm sisi. Energi E(G) dari G, dinotasikan dengan E(G), didefinisikan sebagai jumlahan dari nilai mutlak dari nilai-nilai eigenG. Pada paper ini, kami menyatakan dua batas atas baru untuk energi dari graf, satu batas dalam sukum,ndan batas yang lain dalam suku nilai eigen mutlak terbesar dan terkecil. Paper ini juga memuat batas atas untuk energi Laplace dari graf.
Kata kunci: Matriks ketetanggaan, matriks Laplace, energi dari graf, energi Laplace dari graf.
2000 Mathematics Subject Classification: Primary 05C50, 05C69. Received: 26 Sept 2016, revised: 25 March 2017, accepted: 26 March 2017.
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1. Introduction
The concept of energy of a graph was introduced by I. Gutman [6] in the year 1978. LetGbe a graph withnvertices{v1, v2, ..., vn} andmedges and A= (aij)
be the adjacency matrix of the graph. The eigenvaluesλ1, λ2,· · ·, λnof A, assumed
in non increasing order, are the eigenvalues of the graphG. The energyE(G) ofG
is defined to be the sum of the absolute values of the eigenvalues ofG. i.e.,E(G) =
n
X
i=1
|λi|. For details on the mathematical aspects of the theory of graph energy see
the papers [2, 3, 8] and the references cited there in. The basic properties including various upper and lower bounds for energy of a graph have been established in [10] and it has found remarkable chemical applications in the molecular orbital theory of conjugated molecules [5, 9]. The bounds for eigenvalues of graph can be found in [1,13].
Definition 1.1. Let G be a graph with n vertices and m edges. TheLaplacian matrixof the graph G, denoted byL= (Lij), is a square matrix of order n whose elements are defined as
Lij =
−1 if vi andvj are adjacent
0 if vi andvj are not adjacent di if i= j
wheredi is the degree of the vertexvi. Eigenvalues ofL is called eigenvalues ofG.
Definition 1.2. Letµ1, µ2,· · ·, µn be the Laplacian eigenvalues of G.Laplacian
energyLE(G) of G is defined as LE(G) = n
X
i=1
µi−
2m n
.
The matrixL is positive semi-definite and therefore its eigenvalues are non-negative. The least eigenvalue is always equal to zero. The second largest eigenvalue is called the algebraic connectivity ofG. The basic properties including various up-per and lower bounds for Laplacian energy have been established in [7, 11, 12, 13].
2. Main Results
2.1. Energy of graph. We denote the decreasing order of the the absolute value of eigenvalues ofGbyρ1≥ρ2≥...≥ρn. The following are the elementary results
that follows from this notation. (1)ρi=|λk|for some k
(2)ρi≥λi for all i
(3)E(G) =
n
X
(4)ρn≤ n
X
i=1
ρi=E(G)
(5) By Cauchy-Schwarz inequality
Xn
(6) Let GandH be any two graphs with same nvertices each. Let their number of edges be respectivelym1 and m2. Ifρ1≥ρ2 ≥...≥ρn andρ
are their the absolute value of eigenvalues then
n
By Cauchy Schwarz inequality
24
2m−ρ2n≥
1 (n−1)
nX−1
i=1 ρi
2
≥ 1
(n−1) ρ
2 n
⇒ρn≤
r
2m(n−1)
n
which is an upper bound for the smallest absolute eigenvalue of the graphG
Theorem 2.2. Let Gbe a graph withnvertices andm edges. Letρ1≥ρ2≥...≥ ρn be the the absolute value of eigenvalues of G. If ρ1 is repeatedk times then
ρ1≤
1
k(p−1)
p
2mkp(p−1)−
kp
X
i=k+1 ρi
wherekp≤nandp6= 1,k6= 0.
Proof. LetH = [
k Kp
∪Kn−kp
c
wherekp≤n
That isH is the union of graphsKp, repeatedktimes and a graph (Kn−kp)
c.
The number of vertices ofH isnand the number of edges is kp(p−1) 2 . Its the absolute value of eigenvalues spectrum is
p−1 1 0
k k(p−1) (n−kp)
.
By Cauchy Schwarz inequality
ρ1(p−1)+...+ρk(p−1)+ρk+1(1)+...+ρkp(1)+ρkp+1(0)+...+ρn(0)≤2
q
mkp(p−1)
2
Butρ1=ρ2=...=ρk
∴(p−1)kρ1+
kp
X
i=k+1 ρi≤2
r
mkp(p−1)
2
ρ1≤
1
k(p−1)
p
2mkp(p−1)−
kp
X
i=k+1 ρi
. Here (p6= 1, k6= 0)
Corollary 2.3. If kp=n, then by the above theorem
(n−k)ρ1+ n
X
i=k+1 ρi≤
r
2mn(n−k)
k
(n−k)ρ1+E(G)−kρ1≤
r
2mn(n−k)
(n−2k)ρ1+E(G)≤
r
2mn(n−k)
k
E(G)≤ r
2mn(n−k)
k −(n−2k)ρ1
Also ifp= 2 and2k=nthen the upper bound for energy of graph is
E(G)≤ r
2mn(2k−k)
k
E(G)≤√2mn.
Corollary 2.4. If kp=n−1, then we get the following result.
E(G)−ρn≤
r
2m(n−1)(n−1−k)
k −(n−1−2k)ρ1
E(G)≤ r
2m(n−1)(n−1−k)
k −(n−1−2k)ρ1+ρn.
Also ifp= 2 and2k=n−1 then the upper bound for energy of graph is E(G)≤p2m(n−1) +ρn.
Corollary 2.5. If k= 1, then E(G)≤p
2mn(n−1)−(n−2)ρ1 forp=n. andE(G)≤p
2m(n−1)(n−2)−(n−3)ρ1+ρn forp=n−1.
Corollary 2.6. Since ρ1≥ 2 m
n andρn ≤
r
2m
n we get new upper bound for energy of graph in term ofm andn
E(G)≤ r
2mn(n−k)
k −(n−2k)
2m
n forpk=n.
E(G)≤ r
2m(n−1)(n−1−k)
k −(n−1−2k)
2m n +
r
2m
n forpk=n−1.
Corollary 2.7. For a r-regular graph m= rn
2 andρ1 =r we have the following upper bound
E(G)≤n
q
r(n−k)
k −(n−2k)rforpk=n.
E(G)≤qrn(n−1)(n−1−k)
k −(n−1−2k)r+
√
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Theorem 2.8. Let Gbe a graph withnvertices andm edges. Letρ1≥ρ2≥...≥ ρn be the the absolute value of eigenvalues of G. If ρ1 is repeatedk times then
ρ1≤
1
k
2√mk−
2k
X
i=k+1 ρi
. (k6= 0)
Proof. Here we compare the absolute value of eigenvalues ofGwith absolute eigen-value of the graphH = [
k Kp,q
.
Selectpandqsuch thatn=k(p+q). The number of vertices ofH isnand the number of edges iskpq. Its the absolute value of eigenvalues spectrum are
√
pq 0 2k (n−2k)
.
By Cauchy Schwarz inequality
ρ1√pq+...+ρk√pq+ρk+1√pq+...+ρ2k√pq+ρ2k+1(0) +...+ρn(0)≤2√mkpq
Butρ1=ρ2=...=ρk
∴ ρ1k√pq+√pq
2k
X
i=k+1 ρi≤2
p
mkpq
ρ1k+ 2k
X
i=k+1 ρi≤2
√
mk
ρ1≤
1
k
2√mk−
2k
X
i=k+1 ρi
.
Corollary 2.9. If p=q= 1and2k=nthen
ρ1k+ n
X
i=k+1 ρi≤2
r
mn
2
i.e.,E(G)≤√2mn.
Corollary 2.10. Ifp=q= 1 and2k=n−1then
ρ1k+ n−1
X
i=k+1 ρi≤2
r
m(n−1)
2
⇒E(G)- ρn≤
p
2m(n−1)
i.e.,E(G)≤p
i.e.,E(G)≤p
2m(n−1) +q2mn .
Corollary 2.11. Fork= 1 ,ρ1+ρ2≤2√m.
Using the above corollary we obtain another bound for energy of graphs.
Theorem 2.12. Let Gbe a graph withnvertices andm edges and2m≥n. If the first absolute eigenvalue,ρ1not repeated thenE(G)≤√m(2 +√2n−4)
Proof. Cauchy Schwarz inequality for (n−2) terms is
Xn
i=3 aibi
2 ≤
n
X
i=3 a2i
Xn
i=3 b2i
Putai=ρi andbi= 1 n
X
i=3 ρi≤
v u u t
Xn
i=3 ρ2i
Xn
i=3
1
E(G)−(ρ1+ρ2)≤
p
(2m−(ρ2
1+ρ22))(n−2)
E(G)≤(ρ1+ρ2) +√n−2
p
(2m−(ρ2 1+ρ22))
Butρ1+ρ2≤2√m ∴ E(G)≤2√m+√n−2
p
(2m−(ρ2 1+ρ22))
We maximize the functionf(x, y) = 2√m+√n−2p
(2m−(x2+y2))
Thenfx= −
√
n−2x
p
(2m−(x2+y2)) andfy =
−√n−2y
p
(2m−(x2+y2))
For maxima valuefx= 0 andfy = 0 which implies (x, y)≡(0,0)
fxx= −
√
n−2(2m−y2)
(2m−(x2+y2))3 2
, fyy =−
√
n−2(2m−x2)
(2m−(x2+y2))3 2
,fxy=
√n −2xy
(2m−(x2+y2)) 3 2
At (x, y)≡(0,0),fxx=−
q
n−2
2m, fyy =−
q
n−2
2m,fxy= 0 and
∆ =fxxfyy−(fxy)2= n2m−2
Thusf(x, y) attains maximum value at (0,0)∴ f(0,0) =√m(2 +√2n−4)
E(G)≤√m(2 +√2n−4).
2.2. Laplacian energy of graph. Analogous to the bounds for energy of graphs, now we obtain bounds for Laplacian energy of graphs.
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Proof. By Cauchy Schwarz inequality
n ...≥σn be the absolute Laplacian eigenvalues ofG. Ifσ1 is repeatedktimes then
σ1≤ 1 Its the absolute value of eigenvalues spectrum is
p−1 1 0
k k(p−1) (n−kp)
.
By Cauchy Schwarz inequality
kσ1≤
Corollary 2.15. Ifkp=nthen by the above theorem
(n−k)σ1+
Corollary 2.16. Ifkp=n−1 we get the following result.
LE(G)−σn≤ Laplacian energy of graph
LE(G)≤
Corollary 2.17. Ifk= 1then the upper bounds changes to
30 ...≥σn be the absolute Laplacian eigenvalues ofG. Ifσ1 is repeatedktimes then
σ1≤ 1
Proof. Here we compare absolute Laplacian eigenvalues ofGwith absolute eigen-value of graphH = [ the number of edges iskpq. Its the absolute value of eigenvalues spectrum is
√
pq 0
2k (n−2k)
.
By Cauchy Schwarz inequality
σ1√pq+...+σk√pq+σk+1√pq+...+σ2k√pq+σ2k+1(0)+...+σn(0)≤
Corollary 2.19. If2k=nthen by above theorem
LE(G)≤p
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