On eccentricity version of Laplacian energy of a graph

0.06 MB·

by  Nilanjan De

On eccentricity version of Laplacian energy of a graph
NilanjanDe
∗
DepartmentofBasicSciencesandHumanities(Mathematics),
CalcuttaInstituteofEngineeringandManagement,Kolkata,India.
7
1
0
2

n
a
J Abstract

8
Theenergy ofagraphG isequal to thesum ofabsolutevalues oftheeigenvalues

] oftheadjacency matrixofG, whereas theLaplacian energy of agraphG is equal
M
to the sum of the absolute value of the difference between the eigenvalues of the
D
Laplacian matrix of G and average degree of the vertices of G. Motivated by
.
s the work from Sharafdini et al. [R. Sharafdini, H. Panahbar, Vertex weighted
c
[ Laplacian graphenergy andothertopologicalindices. J. Math. Nanosci. 2016,6,

49-57.], in this paper we investigate the eccentricity version of Laplacian energy
1
v ofagraphG.
0
0 MSC(2010):Primary: 05C05.
0
2
0
Keywords: Eccentricity;Eigenvalue;Energy (ofgraph); Laplacian energy;
.
1 Topologicalindex.
0
7
1
: 1. Introduction
v
i
X
LetG be a simplegraph withn vertices and m edges. Let thevertex and edge
r
a sets of G are denoted by V(G) and E(G) respectively. The degree of a vertex v,
denoted by d (v), is the number of vertices adjacent to v. For any two vertices
G
u,v V(G), the distance between u and v is denoted by d (u,v) and is given by
G
∈
the number of edges in the shortest path connecting u and v. Also we denote the
sum of distances between v V(G) and every other vertices in G by d(xG), i.e.,
∈ |
d(xG) = d (x,v). The eccentricity of a vertex v, denoted by ε (v), is
v V(G) G G
| ∈
the largest distance from v to any other vertex u of G. The total eccentricity of a
P
CorrespondingAuthor.
∗
Emailaddress: [email protected](NilanjanDe)
graph is denoted by ζ(G) and is equal to sum of eccentricities of all the vertices
of the graph. Let A = [a ] be the adjacency matrix of G and let λ ,λ ,...,λ are
ij 1 2 n
eigenvaluesof A whicharetheeigenvaluesofthegraphG. Theenergyofagraph
is introduced by Ivan Gutman in 1978 [1] and defined as the sum of the absolute
valuesofitseigenvaluesand is denotedbyE(G). Thus
n
E(G) = λ .
i
| |
i=1
X
Alargenumberofresultsonthegraphenergyhavebeenreported,seeforinstance
[4,5,6,7]. Motivatedbythesuccessofthetheoryofgraphenergy,otherdifferent
energylikequantitieshavebeenproposedandstudiedbydifferentresearcher. Let
D(G) = [d ] be the diagonal matrix associated with the graph G, where d =
ij ii
d (v) and d = 0 if i , j. Define L(G) = D(G) A(G), where L(G) is called the
G i ij
−
Laplacian matrix ofG. Let µ ,µ ,...,µ be the Laplacian eigenvalues ofG. Then
1 2 n
theLaplacian energy ofG isdefined as [2]
n
2m
LE(G) = µ .
i
| − n |
i=1
X
Various study on Laplacian energy of graphs were reported in the literature [8,
9, 10, 11, 12]. Analogues to Laplacian energy of a graph a different new type of
graph energy were introduced and in this present study, inspired by the work in
[3],weinvestigatetheeccentricityversionofLaplacianenergyofagraphdenoted
by LE (G). In this case, we define the Laplacian eccentricity matrix as L (G) =
ε ε
ε(G) A(G),whereε(G) = [e ]isthe(n n)diagonalmatrixofGwithe = ε (v)
ij ii G i
− ×
and e = 0 if i , j. Here, ε (v) is the eccentricity of the vertex v,i = 1,2,...,n.
ij G i i
Let µ′,µ′,...,µ′ be the eigenvalues of the matrix L (G). Then the eccentricity
1 2 n ε
versionofLaplacian energy ofG isdefined as
n
ζ(G)
LE (G) = µ′ .
ε | i − n |
i=1
X
Recall that ζ(G)/n is the average vertex eccentricity. In this paper, we fist calcu-
late some basic properties and then establish some upper and lower bounds for
EL (G).
ε
2. MainResults
We knowthat,theordinary and Laplacian graph eigenvaluesobeythefollow-
ingrelations:
2
n n
λ = 0; λ2 = 2m
i i
i=1 i=1
n n n
Pµ = 2mP; µ2 = 2m+ d(v)2.
i i i
i=1 i=1 i=1
As tPhe LaplaciaPn spectrum is dPenoted by µ′,µ′,...,µ′, let ν′ = µ′ ζ(G) . Recall
1 2 n i | i − n |
thatthefirstZagrebeccentricityindexofagraphisdenotedby E (G)andisequal
1
to the sum of square of all the vertices of the graph G. Thus, we have E (G) =
1
n
ε(v)2 (for details see [13, 14, 15]). In the following, now we investigatesome
i
i=1
bPasicproperties ofµ′ and ν′.
i i
Lemma 1. Theeigenvaluesµ′,µ′,...,µ′ satisfiesthefollowingrelations
1 2 n
n n
2
(i) µ′ = ζ(G)and (ii) µ′ = E (G)+2m.
i i 1
i=1 i=1
P P
Proof. (i)Since,thetraceofasquarematrixisequaltothesumofitseigenvalues,
wehave
n n
µ′ = tr(EL (G)) = [ε(v) a ] = ζ(G).
i ε i − ii
i=1 i=1
(ii)Againwehave,
P P
n
µ′2 = tr[(ε(G) A(G))(ε(G) A(G))T]
i − −
i=1
X
= tr[(ε(G) A(G))(ε(G)T A(G)T)]
− −
= tr[ε(G)ε(G)T A(G)ε(G)T A(G)Tε(G)+A(G)A(G)T]
− −
n n
= ε(v)2 + λ2
i i
i=1 i=1
X X
= E (G)+2m.
1
Lemma 2. Theeigenvaluesν′,ν′,...,ν′ satisfiesthefollowingrelations
1 2 n
n n
(i) ν′ = 0 and(ii) ν′2 = E (G) ζ(G)2 +2m.
i i 1 − n
i=1 i=1
P P
n n n
Proof.(i)Wehavefrom definition, ν′ = µ′ ζ(G) = µ′ ζ(G) = 0. (cid:3)
i i − n i −
i=1 i=1 i=1
(ii)Again,similarlywehave P P(cid:16) (cid:17) P
n n 2
ζ(G)
2
ν′ = µ′
i i − n
i=1 i=1 !
X X
3
n ζ(G)2 ζ(G)
2
= µ′ + 2µ′
i n2 − i n
i=1 " #
X
ζ(G)2 ζ(G)2
= E (G)+2m+ 2 ,
1
n − n
whichprovesthedesired result. (cid:3)
Lemma 3. Theeigenvaluesν′,ν′,...,ν′ satisfiesthefollowingrelations
1 2 n
1 ζ(G)2
ν′ν′ = E (G) +2m .
i j 2 1 − n
(cid:12) (cid:12)
(cid:12)i<j (cid:12) " #
(cid:12)X (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. Since n ν′ = 0,(cid:12)(cid:12)so wec(cid:12)(cid:12)an write n ν′2 = 2 ν′ν′. Thus,
i i − i j
i=1 i=1 i<j
P P P
n ζ(G)2
2
2 ν′ν′ = ν′ = E (G) +2m.
i j i 1 − n
(cid:12) (cid:12)
(cid:12)i<j (cid:12) i=1
(cid:12)X (cid:12) X
(cid:12) (cid:12)
Hencethedesiredre(cid:12)(cid:12)sult follo(cid:12)(cid:12)ws. (cid:3)
(cid:12) (cid:12)
As an example, in the following, we now calculate the eccentric version of
Laplacian energy and corresponding spectrum of two particular type of graphs,
namely,completegraphand completebipartitegraph.
Example1. Let K bethecompletegraphwithn vertices, then
n
1 1 1 ... 1
− − −
1 1 1 ... 1
 − − − 
L (K ) = 1 1 1 ... 1 .
Itscharacteristicequatioεnins  −−..1. −−..1. −..1. ...... −.1.. 
(n 1)
(µ′ 2) − (µ′ (2 n)) = 0.
− − −
ThustheeccentricityLaplacianspectrumof K isgiven by
n
(1 n) 1
spec (G) = −
ε 1 (n 1)
− !
andhence EL (G) = 2(n 1).
ε
−
4
Example2. Let K be the complete graph with (m+n) vertices and mn edges,
m,n
then
2 0 0 ... 0 1 1 1 ... 1
− − − −
0 2 0 ... 0 1 1 1 ... 1
 − − − − 
0 0 2 ... 0 1 1 1 ... 1
ItscharactLeεr(iKstni,cn)e=quat−−−−i.0.o..1111nis−−−−0....1111 −−−−0....1111 ................... −−−−2....1111 −−..2000..1 −−..0200..1 −−..0020..1 ................... −−..0002..1 .
2(n 1)
(µ′ 2) − (µ′ (2+n))(µ′ (2 n)) = 0.
− − − −
So,theeccentricityLaplacianspectrumof K isgiven by
n,n
n n 0
spec (G) = −
ε 1 1 2(n 1)
− !
andhence EL (K ) = 2n.
ε n,n
Note that, from the above two examples, the properties of the eigenvalues
ν′,ν′,...,ν′ can be verified easily. In the following, next we investigate some
1 2 n
upperand lowerboundsofeccentricityversionofLaplacianenergy ofagraphG.
Theorem 1. Let G beaconnected graphoforder nand sizem, then
1 1
EL (G) 2 m+ (E (G) ζ(G)2).
ε 1
≥ 2 − n
r
n
Proof. Wehavefrom definition, EL (G) = ν′ . So wecan write,
ε | i|
i=1
P
n ζ(G)2
EL (G)2 = ν′2 +2 ν′ν′ E (G) +2m +2 ν′ν′ .
ε i | i j| ≥ 1 − n | i j|
i=1 i<j ! i<j
X X X
HenceusingLemma3, thedesired resultfollows. (cid:3)
5
Theorem 2. LetGbeaconnectedgraphofordernandsizem;andν′ andν′ are
1 n
maximumandminimumabsolutevaluesof ν′s, then
i
n2
ELε(G) ≥ nE1(G)−ζ(G)2 +2mn− 4 (ν′1 −ν′n)2.
r
Proof. Let a and b, 1 i n are non-negative real numbers, then using the
i i
≤ ≤
Ozekisinequality[17], wehave
n n n 2 n2
a2 b2 ab (M M m m )2
i i i i 1 2 1 2
− ≤ 4 −
i=1 i=1 i=1 
where M = maxX(a), mX= min(aX), M = max(b) and m = min(b). Let a = ν′
1 i 1 i 2 i 2 i i | i|
and b = 1,then fromabovewehave
i
n n n 2 n2
ν′ 2 12 ν′ (ν′ ν′)2.
| i| − | i| ≤ 4 1 − n
i=1 i=1 i=1 
Thus,wecan wrXite X X 
n n2
EL (G)2 n ν′ 2 (ν′ ν′)2
ε ≥ | i| − 4 1 − n
i=1
X
ζ(G)2 n2
2
= n E (G) +2m (ν′ ν′) ,
1 − n − 4 1 − n
!
fromwhere thedesired resultfollows. (cid:3)
Theorem 3. LetGbeaconnectedgraphofordernandsizem;andν′ andν′ are
1 n
maximumandminimumabsolutevaluesof ν′s, then
i
1 ζ(G)2
EL (G) E (G) +2m nν′ν′ .
ε ≥ (ν′1 +ν′n) " 1 − n − 1 n#
Proof. Let a and b, 1 i n are non-negative real numbers, then from Diaz-
i i
≤ ≤
Metcalfinequality[16], wehave
n n n
b2 +mM a2 (m+ M) ab .
i i i i
≤
i=1 i=1 i=1 
X X X 
6
where, ma b Ma. Let a = 1 and b = ν′ , then fromabovewehave
i ≤ i ≤ i i i | i|
n n n
ν′ 2 +ν′ν′ 12 (ν′ +ν′) ν′ .
| i| 1 n ≤ 1 n | i|
i=1 i=1 i=1 
Now,since X X X 
n
EL (G) = ν′
ε | i|
i=1
X
n
and ν′2 = E (G) ζ(G)2 +2m,weget
i 1 − n
i=1
P
ζ(G)2
E (G) +2m+nν′ν′ (ν′ +ν′)EL (G),
1 − n 1 n ≤ 1 n ε
fromwhere thedesired resultfollows. (cid:3)
Theorem 4. Let G beaconnected graphoforder nand sizem, then
EL (G) nE (G) ζ(G)2 +2mn .
ε 1
≤ −
q
h i
Proof. Using the Cauchy-Schwarz inequality to the vectors (ν′ , ν′ ,..., ν′ ) and
| 1| | 2| | n|
(1,1,...,1),wehave
n n
ν′ √n ν′ 2
| i| ≤ vt | i|
i=1 i=1
X X
Thusfrom definition,wehave
n n
ELε(G) = |ν′i| ≤ √nvt |ν′i|2
i=1 i=1
X X
n ζ(G) ζ(G)2
= √nvti=1 |µ′i − n |2 = √ns"E1(G)− n +2m#.
X
Hencethedesiredresult follows. (cid:3)
Theorem 5. Let G beaconnected graphoforder nand sizem, then
2 ν′1ν′n
EL (G) nE (G) ζ(G)2 +2mn .
ε ≥ ν′1q+ν′n q 1 −
h i
7
Proof. We have, from Polya-Szego inequality [18] for non-negatve real numbers
a and b, 1 i n
i i
≤ ≤
n n 2 n 2
1 M M m m
a2 b2 1 2 + 1 2 ab
i i i i
≤ 4 m m M M
i=1 i=1 r 1 2 r 1 2 i=1 
where M = mXax(a),Xm = min(a), M = max(b) andmX= min(b). Let a = ν′
1 i 1 i 2 i 2 i i | i|
and b = 1,then fromabovewehave
i
2
n ν′ 2 n 12 1 ν′n + ν′1 n ν′ 2.
i=1 | i| i=1 ≤ 4sν′1 sν′n i=1 | i|
Thatis, X X   X 
2
n n ν′ 2 1 ν′n + ν′1 (EL (G))2
Xi=1 | i| =≤ 14(νs′n +ν′1ν′1)2s(EνL′n(G))2ε.
ε
4 ν′nν′1
n
Since, ν′2 = E (G) ζ(G)2 +2m, wegetthedesired resultfrom above. (cid:3)
i 1 − n
i=1
P
3. Conclusion
In this paper, we study different properties and boundsof eccentricity version
of Laplacian energy of a graphG. It is found that, there is great analogy between
theoriginalLaplacianenergyandeccentricityversionofLaplacianenergy,where
as alsohavesomedistinctdifference.
Competing Interests
Theauthordeclares thatthereis noconflict ofinterestsregardingthepublica-
tionofthispaper.
8
References
[1] I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz.
Graz, 103(1978)1-22.
[2] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl.
414 (2006)29–37.
[3] R. Sharafdini, H. Panahbar, Vertex weighted Laplacian graph energy and
othertopologicalindices,J. Math.Nanosci.6 (2016)49-57.
[4] V. Nikiforov,The energy ofgraphs and matrices, J. Math. Anal. Appl. 326
(2007)1472-1475.
[5] R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004)
287–295.
[6] I. Gutman, The energy of graph: Old and new results, Algebraic combina-
torics andapplications,Springer, Berlin, (2001)196-211.
[7] H. Liu, M. Lu and F. Tian, Someupperbounds forthe energy of graphs, J.
Math.Chem., 41(1)(2007)45-57.
[8] G. H. Fath-Tabar, A.R. Ashrafi, Some remarks on Laplacian eigenvalues
and Laplacian energy ofgraphs,Math.Commun.,15(2)(2010)443–451.
[9] K.C. Das, S.A. Mojallala and I. Gutman, On energy and Laplacian energy
ofbipartitegraphs, Appl.Math.Comput.273(2016)759-766.
[10] B. Zhou, I. Gutman and T. Aleksic, A note on Laplacian energy of graphs,
MATCH Commun.Math.Comput.Chem. 60(2008)441-446.
[11] T. Aleksic, Upper bounds for Laplacian energy of graphs, MATCH Com-
mun,Math.Comput.Chem. 60 2008435-439.
[12] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl.
197-198(1994)143-176.
[13] R. Xing, B. Zhou and N. Trinajstic, On Zagreb eccentricity indices, Croat.
Chem. Acta, 84(4)(2011)493–497.
9
[14] Z. Luo and J. Wu, Zagreb eccentricity indices of the generalized hierar-
chical product graphs and their applications, J. Appl. Math. 2014, DOI
10.1155/2014/241712.
[15] N.De,NewboundsforZagrebeccentricityindices,OpenJ.DiscreteMath.
3 (2013)70–74.
[16] J.B.Diaz,F.T.Metcalf,StrongerformsofaclassofinequalitiesofG.Plya-
G.Szego and L. V. Kantorovich. Bulletin of the AMS - American Mathe-
matical Society,69(1963)415–418.
[17] N. Ozeki, On the estimation of inequalities by maximum and minimum
values,J. CollegeArts Sci. ChibaUniv.5(1968)199–203,in Japanese.
[18] G. Polya, G. Szego, Problems and Theorems in analysis, Series, Integral
Calculus, TheoryofFunctions,Springer, Berlin, (1972).
10