1

The Eigen-complete Difference Ratio of classes of Graphs- Domination,Asymptotes and Area

Paul August Winter* and Samson Ojako Dickson

AbstractThe energy of a graph is related to the sum of -electron energy in a moleculerepresented by a molecular graph and originated by the HMO (Hückel molecular orbital)theory. Advances to this theory have taken place which includes the difference of theenergy of graphs and the energy formation difference between and graph and itsdecomposable parts. Although the complete graph does not have the highest energy of allgraphs, it is significant in terms of its easily accessible graph theoretical properties and hasa high level of connectivity and robustness, for example. In this paper we introduce a ratio,the eigen-complete difference ratio, involving the difference in energy between thecomplete graph and any other connected graph G, which allows for the investigation of theeffect of energy of G with respect to the complete graph when a large number of verticesare involved. This is referred to as the eigen-complete difference domination effect. Thisdomination effect is greatest negatively (positively), for a strongly regular graph (stargraphs with rays of length one), respectively, and zero for the lollipop graph. When thisratio is a function f(n), of the order of a graph, we attach the average degree of G to theRiemann integral to investigate the eigen-complete difference area aspect of classes ofgraphs. We applied these eigen-complete aspects to complements of classes of graphs.AMS Classification: 05C50*Corresponding author: email: winterp@ukzn.ac.zaKey words: Graph energy, energy difference between graphs, ratios, domination,asymptotes, areas

2 1. IntroductionIn this paper graphs G will be on n vertices. We shall adopt the definitions andnotation of Harris, Hirst, and Mossinghoff. It is assumed that G is simple, thatis, it does not contain loops or parallel edges.The energy of a graph is the sum of the absolute values of the eigenvalues ofthe adjacency matrix of the graph in consideration. This quantity is studied inthe context of spectral graph theory. In short, for an n -vertex graph G withadjacency matrix A having eigenvalues n 21 , the energy )(GE isdefined as:

n

iiGE

1

It is related to the sum of -electron energy in a molecule represented by amolecular graph. If we know some chemistry, then we might fully appreciatethe origin of graph energy. In a private communication, Gutman (see Gutman)claimed that the HMO (Hückel molecular orbital) theory is nowadayssuperseded by new theories that provide better explanations and which donot make unnecessary assumptions.Graph energy became a very popular topic of mathematical research; this isevident in the reviews and recent papers.In the paper “Energy of Graphs” by Brualdi the difference of the energy of twographs G and H on the same number n of vertices was presented.Although the complete graph nK does not have the maximum energy of allgraphs (see Haemers), it is a very important and well-studied class of graphs –for example it has a high degree of connectiveness and robustness. Thus onewould like to compare its energy with the energy of any other graph G interms of how close their energies are, and how the energy of G compares withthe energy of nK where a large number of vertices are involved. This energy

3idea can be translated to that of molecules made up of atoms with bonds,where we map the atoms to vertices and bonds to edges, and the dominationeffect will allow for the investigation of how other molecular energiescompare with that of a molecule with all possible bonds between atoms.The eigen-complete difference ratio allowed for the investigation of thedomination effect of the energy of graphs on the energy of the complete graphwhen a large number of vertices are involved. We found that this dominationeffect is the greatest negatively (positively) for a strongly regular graph (stargraphs with rays of length one), respectively. and is zero for the lollipopgraph.Ratios and graphs

Ratios have been an important aspect of graph theoretical definitions.Examples of ratios are: expanders, (see Alon and Spencer ), the central ratioof a graph (see Buckley), eigen-pair ratio of classes of graphs (see Winter andJessop), Independence and Hall ratios (see Gábor), tree-cover ratio of graphs(see Winter and Adewusi), eigen-energy formation ratio of graphs (seeWinter and Sarvate), t-compete sequence ratio (see Winter, Jessop andAdewusi) and the chromatic-cover ratio of graphs (see Winter).We now introduce the idea of ratio, asymptotes and areas involving energydifference between the complete graph and G, similar to that of Winter andAdewusi, Winter and Jessop, Winter and Sarvate, Winter, Jessop andAdewusi, and Winter.

2. Eigen-complete difference ratio- asymptotes, domination effect and areaLet nK be the complete graph on n vertices.Definition 2.1

4

The difference between the energy of nK and a connected graph G on thesame number of vertices n is given by:)()( GEKED n

Gn And is called the eigen-complete difference associated with G.If the graph G in belongs to a class of graphs of order n, then the complete-

energy difference associated with is defined as:

GGEKED nn );()( .Dividing the complete-energy difference by the energy of nK will give an“average” of the complete-energy difference with respect to G. This providesmotivation for the following definition:Definition 2.2The eigen-complete difference ratio with respect to )(G , respectively, isdefined as:)(

)()(

n

nGn KE

GEKEDRat

;

G

KE

GEKEDRat

n

nn ;

)(

)()(

Definition 2.3If the eigen-complete difference ratio is a function f(n) of the order of G ,then its horizontal asymptote results in the eigen-complete differenceasymptote:

GKE

GEKELimDAsymrat

n

n

nn ];

)(

)()([

This asymptote allows for the investigation of the effect of the energy of agraph G on the complete graph when a large number of vertices are involved,referred to as the domination eigen-complete difference effect.

5Definition 2.4Attaching the average degree of graph G, with m’ edges, to the Riemannintegral of

GKE

GEKEDRat

n

nn ;

)(

)()( we obtain the eigen-complete

difference area:

dnKE

GEKE

n

mDArat

n

nn ]

)(

)()([

'2 ; with 0kDArat where k is the smallestorder of G .The average degree is referred to as the length of the area, while the integralpart is the height of the area.LemmaThe eigen-complete difference ratio can take on one of the following:(1)

G

KE

GE

KE

GEKEDRatKEEG

nn

nnn ;0

)(

)(1

)(

)()()()

(2)

GKE

GE

KE

GEKEDRatKEEG

nn

nnn ;0

)(

)(1

)(

)()()()

(3)

GKE

GEKEDRatKEEG

n

nnn ;0

)(

)()()()

6 3. Examples3.1The complete split-bipartite graph

2,

2

nnK .The energy of this graph is n and it has

4

2n edges while that of the completegraph is 2n-2 so that:22

2

22

2

22

)22(

22

)()(2

,2

n

n

n

n

n

nn

n

KEKE

DRatnnn

n

2

1]

22

2[

n

nLimDAsymratn

n

))1ln((41

2

4]

22

2[

2cnn

ndn

n

nndn

n

nnDArat n with smallest order2 we have :

2c

3.2 The star graph 1,1 nK with n-1 rays of length1.The energy of this star graph is 12 n so that:

1

11

)1(2

12)22(

22

)()( 1,1

nn

nn

n

KEKEDRat nn

n

7

1]1

11[

nLimDAsymratn

n

]12[)1(2

]1

11[

)1(2cnn

n

ndn

nn

nDArat n

With smallest star graph on 2 vertices we have:0c .

3.3 Star graphs 2,rS with r rays of length 2The energy of this star graph with 1 nr edges is:123 nn so that:

22

121

22

)123()22(

22

)()( 1,1

n

nn

n

nnn

n

KEKEDRat nn

n

2

1]

22

121[

n

nnLimDAsymratn

n

dnn

n

nn

n

n

ndn

n

nn

n

nDArat n ]

1

1

2

1

)1(2

2

)1(2

1[

)1(2]

22

121[

)1(2

].2

1)1ln(

2[

)1(2cAn

n

n

n

dnn

nA

1

1 let uduu

uAududnun 2

221

22

.

8

Put tu tan2 so that ]2

)tanln(sectansec[4sec4 3 tttt

tdtA

Thus: )2

1

2

1ln(21)]

2

1

2

1ln(

2

1

2

1[2 2

nn

nnnnn

A

The smallest such star graph is non 3 vertices so that)52ln(28))

2

810ln(28

A

Thus:2ln

2

3)52ln(22 cAnd eigen-complete difference area is:

]2ln2

3)52ln(22)]

2

1

2

1ln(21.[

2

1)1ln(

2[

)1(2 2

nn

nnn

n

n

3.4 The line graph of nK

The line graph )( nKL of nK has2

)1(

nnp vertices and energy nn 62 2 (see Brualdi). The number q of edges is the sum of the square of the degreesminus the number of edges of nK :

2

)2)(1(]11[

2

)1(

2

)1(

2

)1( 2

nnnn

nnnnnnq

npnnn

nn 4442

)1(462 2

2

811

2

811022 pp

npnn

9Thus:ppn

nnnnKLE n 812244

2

)1(462))(( 2

22

812

22

812422

)(

))(()(

p

pp

p

ppp

KE

KLEKEDRat

p

npp

1]22

812[

p

ppLimDAsymratn

n

8

1;

481;]

22

812

22

22[

2

22

8122

22

up

ududppudp

p

p

p

p

p

qn

signabsoluteomitdpp

pp

p

qDArat n

]9

2)[(

2]

44

9)2(

)[(2

]4

)24

1(

)2()[(

22

2

22 duu

up

p

qudu

u

up

p

qudu

u

up

p

q

cppppp

q

signabsolutereplacingduu

dupp

q

)]381ln(6

7)381(ln(

6

781[

2

]9

7)[(

22

3p yields:)8ln(

6

7)2(ln(

6

72)8ln(

6

7)2(ln(

6

753 c

So that the eigen-complete area of the line graph of nK on p vertices is:)]8ln(

6

7)2(ln(

6

72)381ln(

6

7)381(ln(

6

781[

2 pppp

p

q

10

3.5 Strongly regular graphsKoolen and Moulton have proved that the energy of a graph on n vertices is atmost n(1+ n )/2, and that equality holds if and only if the graph is stronglyregular with parameters (n,(n+ n )/2,(n+2 n )/4,(n+2 n )/4). Such graphsare equivalent to a certain type of Hadamard matrices. Here we surveyconstructions of these Hadamard matrices and the related strongly regulargraphs (see Haemers).Its energy is

2

)1( nn and to find the number of edges m’ we use:4

)(''2

2

)('2)(

1

nnnmm

nnnmvd

n

Thus44

43

222

)1()22(

22

))(()(

n

nnn

n

nnn

n

GSREKEDRat n

n.

]44

43[

n

nnnLimDAsymratn

n

]44

1

4

3[

2

)()]

44

1()

44

44(

4

3[

2

)(

]44

43[

2

)(]

44

43[

'2

dnn

nnn

nndn

n

nn

n

nnn

absoluteremovingdnn

nnnnndn

n

nnn

n

mDArat n

11Nowudu

u

uununputdn

n

nnndn

n

nn2

14

1)1ln(

4

1:

14

1)1ln(

4

1

44

12

22

But duuu

uudu

u

u

u

uudu

u

u

1

1

1

1

2

1

6

1

11

1

2

1

12

122

23

2

2

2

22

2

4

1

1ln

2

1

2

1

6

1

1

1ln

2

1

2

1

6

1 2

33

n

nnn

u

uuu

So the eigen-complete area (without absolute sign) is:]

1

1ln

2

1

2

1

6

1)1ln(

4

1

4

3[

2

)( 2

3

cn

nnnnn

nn

The term 2

3

n dominates for large n, so introducing absolute sign:]

1

1ln

2

1

2

1)1ln(

4

1

4

3

6

1[

2

)( 2

3

cn

nnnnn

nn

3.6 Lollipop graphThe proof of the following theorem can be found in Haemers, Liu and Zhang:Theorem 1Let G be a graph with an end vertex 1x adjacent to vertex 2x , and let 'G be thesubgraph of G induced by removing the vertex 1x . and let ''G be the subgraphof G induced by removing the vertex 2x . Then:

)()()( )''()'()( GAGAGA PPP

12Where )()( GAP is the characteristic polynomial ))(det( IGA , and )(GA theadjacency matrix of G .Example with complete graph joined to end vertexSo if LP(G) is the complete graph on n-1vertices (the base of the lollipopgraph) joined to a single end vertex 2x by an edge 21xx , we have:

))3(()1())2(()1()()()( 32)''()'()( nnPPP nn

GAGAGA

)]3(())2()(1[()1( 3 nnn

)]3()2()2([)1( 23 nnnn

]1)2([()1( 23 nn Roots of quadratic are:2

444)2( 2

nnn ; we have roots:

2

84)2(;

2

84)2();3(1;0

22

nnnnnnncymultipliti

Energy of this graph is therefore:2

)2(84

2

84)2()3(10

22

nnnnnnn since 4n

84)3( 2 nnn

13Theorem 2The energy of the lollipop graph with base the complete graph on n-1 verticesis:84)3( 2 nnn

To find dv for this graph we have:'243)2)(2(1)1()2)(2( 2 mnnnnnnnndv Thus:

22

841

22

]84)3[()22(

22

))(()( 22

n

nnn

n

nnnn

n

GLPEKEDRat n

n

22

841 2

n

nnnLimDAsymratn

n

Multiply top and bottom of22

841 2

n

nnn by:841 2 nnn :

)841)(22(

)84()1(2

22

nnnn

nnn

For large n, the numerator is of order 6n and the denominator is of order 24nso that 04

62

n

nLimDAsymratn

n .

14

dnn

nnn

n

mdn

n

nnn

n

mDArat n 1

841'

22

841'2 22

dnn

nn

n

mdn

n

nn

n

n

n

m

1

4)2(2[

']

1

842

1

1[

' 22

Theorem 3nDRat ;

nDAsymrat and nDArat for the following classes of graphs are,respectively:

2,

2

nnK :22

2

n

n ; )2)1ln((4

;2

1 nn

n

1,1 nK : 1;1

11

n; ]12[

)1(2

nn

n

n

rK ,2 :22

121

n

nn ; ;2

1

]2ln2

3)52ln(22)]

2

1

2

1ln(21.[

2

1)1ln(

2[

)1(2 2

nn

nnn

n

n

)( nKL :.22

812

p

pp ; 1 ;)]8ln(

6

7)2(ln(

6

72)381ln(

6

7)381(ln(

6

781[

2 pppp

p

q

Where2

)2)(1(

nnnq

)(GSR :44

43

n

nnn ; ;]

1

1ln

2

1

2

1)1ln(

4

1

4

3

6

1[

2

)( 2

3

cn

nnnnn

nn

15

)(GLP :22

841 2

n

nnn ;0 ;

dnn

nnn

n

mDArat n 22

841'2

2

Lemma 11

44

43

nDRatn

nnn

ProofSince 0)( GE for any graph G we get the right hand inequality:1

22

)(

22

)()(

n

KE

n

GEKEDRat nn

n

And since2

)1(22))(()())(()(

nnnGSREKEGSREGE n

we getthe left hand inequality.Lemma 2The domination eigen-complete effect is at most one and is greatest negativelyfor the strongly regular graph examined in the above example.ProofThe strongly regular graph in the above example has the greatest energy of allgraphs so that:

]24

43[

n

nnnLimDAsymratn

n

The above lemmas can be used to verify the following theorem:

16Theorem 4]1,(nDAsymrat with end points attained for the strongly regular graphand star graphs with rays of length 1, respectively, and 0nDAsymrat forthe lollipop graph .Corollary 1The eigen-complete difference height of the strongly regular graph above isthe greatest of all eigen-complete heights.ProofSince )'())(( GEGSRE for any graph 'G we have:

dnKE

KEGEdn

KE

KEGSREdn

KE

GSREKE

n

n

n

n

n

n ])(

)()'([]

)(

)()((([]

)(

)((()([

Conjecture 1Except for strongly regular graphs, the eigen-complete difference asymptotelies on the interval [-1,1].4. Eigen-complete different ratios of complements of classes of graphs4.1 The complete-split bipartite graphThe complement of

2,

2

nnK consists of two disjoint copies of,

2

nK . It energy istherefore:42 n so that:

17

1

1

22

2

22

)42(22

22

)()(

nnn

nn

n

GEKEDRat n

n

0]1

1[

nLimDAsymratn

n

])1[ln(2

1

1

2

2cn

n

ndn

nn

nDArat n

Smallest such graph occurs for n=4 so that:3lncThe eigen-complete difference ratio for the complement of the complete-splitbipartite graph is

1

1)(

nnf

The eigen-complete difference ratio of the original graph is:2

)(]1

2)[(')('

2

)('

2

)()(')1

2)(()()(

2

22

2

2222

2

22

22

22

)()()(

nfnnfnf

nnfnfng

nnfnfnf

n

nn

n

n

n

n

nn

n

GEKEDRatng n

n

22

22

)1)(2(

1

)22)(2(

4

)2(

)('2

)(2

1)('

2

)(]

2

2)[('

)22(

2

)22(

)2(2)22()('

nnnnn

ng

nfn

nfnfn

nfnn

nnng

dnn

nfnnIFnn

nfndn

nfd

22 )1(

1)()2()2(;

)1)(2(

1)(

2

1))((

)2()1)(2(

1)(

1

1)()2(

n

c

nnnfc

nnfn

2

)1(

2

11

)2()1)(2(

1

1

1

n

nc

nn

c

nnn

18

1)1(12 cncn

So)2(

1

)1)(2(

1)(

nnnnf

Thus with f(n) and g(n) we associate the quadratic23)( 2 nnnh

)1(222323312)()1( 22 nnnnnnnhnh

)2(2)23312(26344)1()2( 22 nnnnnnnnhnh

)3(22)26344(29396)2()3( 22 nnnnnnnnhnhThe second difference involving (1), (2) and (3) is an arithmetic sequencewith common difference 2. Thus we have the following theorem:Theorem 5The eigen-difference ratio g(n)of the complete-split bipartite and the eigen-difference ratio f(n) its complement are related by the following equation:)2(

)('2)(

2

1)('

n

ngnf

nnf

This results in the differential equation:2)1)(2(

1)(

2

1))((

nnnf

ndn

nfd

With general solution:)2()1)(2(

1)(

n

c

nnnf

Corollary 2The equation in the above theorem yields the following quadratic sequence:

19

,...23,...,6,2,0 2 nnWith second difference sequence with common difference 2:2,4,6,….,2n-2,…4.2 Star graphs with rays of length 1The compliment of the star graph with rays of length one (on at least threevertices) is a complete graph on n-1 vertices together with an isolated vertex.Its energy is therefore:

42 n so that:1

1

22

2

22

4222

22

)()(

nnn

nn

n

GEKEDRat n

n .0

1

1

nLimDAsymratn

n

])1[ln(2

)2)(1(

1

1'cn

n

nndn

nn

mDArat n

.For n=3 we get 2lnc .4.3 The lollipop graph with complete graph on n-1 vertices as baseThe compliment of the lollipop graph consists of a star graphs on n-1 verticesand an isolated vertex. Its energy is therefore 22 n

)(1

21

22

2222

22

)()(ng

n

nn

n

nn

n

GEKEDRat n

n

11

21

n

nnLimDAsymratn

n .

20

dnudunuputdnn

nn

n

mdn

n

nn

n

mDArat n

22];1

2[

'

1

21' 2

.2arctan22

tan2)1(sec2secsec

tan2tan;

1

2

1

2 222

2

2

2

nn

vvdvvvdvv

vvupitdu

u

udn

n

n

Thus:cnnn

n

mDArat n 2arctan22[

' . Taking n=3 we get:1

4

c .

5. ConclusionIn this paper we used the idea of energy difference between two graphs andthe significance of the complete graph to formulate the eigen-completedifference ratio which allowed for the investigation of the domination effectthat the energy of graphs have with respect to the complete graph when alarge number of vertices are involved. This idea can be adopted by moleculeswith a large number of atoms where the need to examine molecules whoseenergy may dominate the molecule that is very well bonded. We found that astrongly regular graph dominated in the largest negative way, while the stargraph with rays of length one had a domination effect of one- the largestpossible positive domination effect. The lollipop graph with base the completegraph had domination effect of zero.We attached the average degree to the Riemann integral of this eigen-complete ratio to determine eigen-complete areas associated with classes ofgraphs and applied the above ideas to the complement of classes of graphs.We showed that the eigen-complete difference ratios of the complete-split

21bipartite graph and its complement are related by a differential equation withan associated quadratic sequence with second difference being a sequencewith common difference of two.6. References

Alon, N. and Spencer, J. H. 2011. Eigenvalues and Expanders. The ProbabilisticMethod (3rd ed.). John Wiley & Sons.Buckley, F. 1982. The central ratio of a graph. Discrete Mathematics. 38(1): 17–21.Brualdi, R. A. 2006. Energy of graphs . Department of Mathematics. Universityof Wisconsin, Madison, WI 53706. brualdi@math.wisc.eduGábor, S. 2006. Asymptotic values of the Hall-ratio for graph powers . DiscreteMathematics.306(19–20): 2593–2601.Gutman I. The energy of a graph, 10. 1978. Steierm arkischesMathema-tisches Symposium (Stift Rein, Graz, 1978),103, 1-22.Harris, J. M., Hirst, J. L. and Mossinghoff, M. 2008. Combinatorics and Graphtheory. Springer, New York.Haemers, W. H. 2008. Strongly regular graphs with maximal energy. LinearAlgebra and its Applications. 429 (11–12), 2719–2723.Haemers, W. H., Liu, X. and Zhang, Y. 2008. Spectral characterizations oflollipop graphs. Linear Algebra and its Applications.428 (11–12), 2415–2423.

22Sarvate, D. and Winter, P. A. 2014. The H-Eigen Energy Formation Number ofH-Decomposable Classes of Graphs- Formation Ratios, Asymptotes and Power.Advances in Mathematics: Scientific Journal. 3 (2), 133_147.Winter, P. A. and Adewusi, F.J. 2014. Tree-cover ratio of graphs withasymptotic convergence identical to the secretary problem. Advances inMathematics: Scientific Journal; Volume 3, issue 2, 47-61.Winter, P. A. and Jessop, C.L. 2014.Integral eigen-pair balanced classes ofgraphs: ratios, asymptotes, areas and involution complementary. To appearin: International Journal of Graph Theory.Winter, P. A. and Sarvate, D. 2014. The h-eigen energy formation number of h-decomposable classes of graphs- formation ratios, asymptotes and power.Advances in Mathematics: Scientific Journal; Volume 3, issue 2, 133-147.Winter, P. A., Jessop, C. L. and Adewusi, F. J. 2015. The complete graph:eigenvalues, trigonometrical unit-equations with associated t-complete-eigensequences, ratios, sums and diagrams. To appear in Journal of Mathematicsand System Science.Winter, P. A. 2015. The Chromatic-Cover Ratio of a Graph: Domination, Areasand Farey Sequences. To appear in the International Journal of MathematicalAnalysis.