Minimum Dominating Color Energy of a Graph

2017·0.15 MB·

by  P.S.K. Reddy

International J.Math. Combin. Vol.3(2017), 22-31
Minimum Dominating Color Energy of a Graph
P.Siva Kota Reddy1, K.N.Prakasha2 and GavirangaiahK3
1. DepartmentofMathematics,SiddagangaInstituteofTechnology,B.H.Road,Tumkur-572103,India
2. DepartmentofMathematic,VidyavardhakaCollegeofEngineering,Mysuru-570002,India
3. DepartmentofMathematics,GovernmentFirstGradeCollge,Tumkur-562102,India
E-mail: [email protected],[email protected],[email protected]
Abstract: Inthispaper,weintroducetheconceptofminimumdominatingcolorenergyof
agraph, ED(G)and computetheminimumdominating color energyED(G)of few families
c c
of graphs. Further, we establish thebounds for minimum dominating color energy.
KeyWords: Minimumdominatingset,Smarandachelydominating,minimumdominating
color eigenvalues, minimum dominating color energy of a graph.
AMS(2010): 05C50
§1. Introduction
Let G be a graph with n vertices v ,v , ,v and m edges. Let A = (a ) be the adjacency
1 2 n ij
···
matrix of the graph. The eigenvalues λ ,λ , ,λ of A, assumedin non increasing order,are
1 2 n
···
the eigenvalues of the graphG. As A is real symmetric, the eigenvalues of G are real with sum
equal to zero. The energy E(G) of G is defined to be the sum of the absolute values of the
eigenvalues of G.
n
E(G)= λ . (1)
i
| |
i=1
X
The concept of graph energy originates from chemistry to estimate the total π-electron
energyofamolecule. Inchemistrythe conjugatedhydrocarbonscanbe representedbyagraph
called molecular graph. Here every carbon atom is represented by a vertex and every carbon-
carbon bond by an edge and hydrogen atoms are ignored. The eigenvalues of the molecular
graph represent the energy level of the electron in the molecule. An interesting quantity in
Hu¨ckel theory is the sum of the energies of all the electrons in a molecule, the so called π-
electron energy of a molecule.
Prof.ChandrashekaraAdigaetal.[5]havedefinedcolorenergyE (G)ofagraphG. Rajesh
c
Khannaetal.[2]havedefinedtheminimumdominatingenergy. Motivatedbythesetwopapers,
we introduced the concept of minimum dominating color energy ED(G) of a graph G and
c
computedminimumdominatingchromaticenergiesofstargraph,completegraph,crowngraph,
and cocktail party graphs. Upper and lower bounds for ED(G) are also established.
c
This paper is organized as follows. In section 3, we define minimum dominating color
energyofagraph. Insection4,minimumdominatingcolorspectrumandminimumdominating
1ReceivedDecember19,2016,Accepted August13,2017.
MinimumDominatingColorEnergyofaGraph 23
colorenergiesarederivedforsomefamiliesofgraphs. Insection5Somepropertiesofminimum
dominatingcolorenergyofagrapharediscussed. Insection6boundsforminimumdominating
color energy of a graph are obtained. section 7 consist some open problems.
§2. Minimum Dominating Energy of a Graph
Let G be a simple graph of order n with vertex set V = v ,v ,v , ,v and edge set E. A
1 2 3 n
···
subsetD V isadominatingsetifD isadominatingsetandeveryvertexofV D isadjacent
⊆ −
toatleastonevertexinD,andgenerally,for O V with O isomorphictoaspecialgraph,
∀ ⊂ h iG
forinstanceatree,aSmarandachelydominatingsetD onO ofGissuchasubsetD V O
S S
⊆ −
thateveryvertexofV D OisadjacenttoatleastonevertexinD . Obviously,ifO = ,D
S S S
− − ∅
is nothing else but the usual dominating set of graph. Any dominating set with minimum car-
dinalityiscalledaminimumdominatingset. LetDbeaminimumdominatingsetofagraphG.
TheminimumdominatingmatrixofGisthen nmatrixdefinedbyA (G)=(a )([2])where
D ij
×
1 if v and v are adjacent,
i j
aij = 1 if i=j and vi D,
 0 otherwise ∈
The characteristic polynomial of A (G) is denoted by f (G,λ) = det(λI A (G)). The
D n D
−
minimum dominating eigenvalues of the graph G are the eigenvalues of A (G).
D
Since A (G) is real and symmetric, its eigenvalues are real numbers and are labelled in
D
non-increasingorderλ λ λ λ The minimumdominatingenergyofGisdefined
1 2 3 n
≥ ≥ ≥···≥
as
n
ED(G)= λ . (2)
c | i|
i=1
X
§3. Coloring and Color Energy
A coloringofgraphG is a coloringof its vertices suchthatno twoadjacentvertices receivethe
samecolor. TheminimumnumberofcolorsneededforcoloringofagraphGiscalledchromatic
number and is denoted by χ(G) ([19]).
Consider the vertex colored graph. Then entries of the matrix A (G) are as follows ([5]):
c
If c(v ) is the color of v , then
i i
1 if v and v are adjacent with c(v )=c(v ),
i j i j
6
aij = 1 if vi and vj are non-adjacent with c(vi)=c(vj),
 −0 otherwise.

The characteristic polynomial of A (G) is denoted by f (G,ρ) = det(ρI A (G)). The
c n c
−
color eigenvalues of the graph G are the eigenvalues of A (G).
c
Since A (G) is real and symmetric, its eigen values are real numbers and are labelled in
c
24 P.SivaKotaReddy,K.N.PrakashaandGavirangaiahK
non-increasing order ρ ρ ρ ρ The color energy of G is defined as
1 2 3 n
≥ ≥ ≥···≥
n
E (G)= ρ . (3)
c i
| |
i=1
X
§4. The Minimum Dominating Color Energy of a Graph
Let G be a simple graph of order n with vertex set V = v ,v ,v , ,v and edge set E. Let
1 2 3 n
···
D be the minimum dominating set of a graph G. The minimum dominating color matrix of G
is the n n matrix defined by AD(G)=(a ) where
× c ij
1 if v and v are adjacent with c(v )=c(v ) or if i=j and v D,
i j i j i
6 ∈
aij = 1 if vi and vj are non adjacent with c(vi)=c(vj),
 −0 otherwise

The characteristic polynomial of AD(G) is denoted by f (G,λ) = det(λI AD(G)). The
c n − c
minimum dominating color eigenvalues of the graph G are the eigenvalues of AD(G).
c
Since AD(G) is real and symmetric, its eigenvalues are real numbers and are labelled in
c
non-increasing order λ λ λ .... λ The minimum dominating color energy of G is
1 2 3 n
≥ ≥ ≥
defined as
n
ED(G)= λ . (4)
c | i|
i=1
X
If the color used is minimum then the energy is called minimum dominating chromatic
energy and it is denoted by ED(G). Note that the trace of AD(G)= D .
χ c | |
§5. Minimum Dominating Color Energy of Some Standard Graphs
Theorem 5.1 If K is the complete graph with n vertices has ED(G)(K ) = (n 2) +
n χ n −
√n2 2n+5.
−
Proof Let K be the complete graph with vertex set V = v ,v , ,v . The minimum
n 1 2 n
{ ··· }
dominating set =D = v .
1
{ }
1 1 1 1 1
···
 1 0 1 1 1 
···
 1 1 0 1 1 
ADc (Kn)= ... ... ... ·..·.· ... ...  .
 
 
 1 1 1 0 1 
 ··· 
 
 1 1 1 1 0 
 ··· n n
  ×
MinimumDominatingColorEnergyofaGraph 25
Its characteristic polynomial is
[λ+1]n−2[λ2 (n 1)λ 1].
− − −
The minimum dominating color eigenvalues are
n 1+√(n2 2n+5) n 1 √(n2 2n+5)
1 − − − − −
spec (K )= − 2 2 .
D n
 
n 2 1 1
−
 
The minimum dominating color energy for complete graph is
(n 1)+ (n2 2n+5)
ED(K ) = 1(n 2)+ − −
χ n |− | − | 2 |
p
(n 1) (n2 2n+5)
+ − − −
| 2 |
p
= (n 2)+ (n2 2n+3),
− −
p
i.e.,
ED(G)(K )=(n 2)+ (n2 2n+5).
χ n − − 2
p
Definition 5.2 The crown graph S0 for an integer n 3 is the graph with vertex set
n ≥
u ,u , ,u ,v ,v , ,v and edge set u v :1 i,j n,i =j . S0 is therefore equiva-
{ 1 2 ··· n 1 2 ··· n} { i i ≤ ≤ 6 } n
lent to the complete bipartite graph K with horizontal edges removed.
n,n
Theorem 5.3 If S0 is a crown graph of order 2n then ED(S0)=(2n 3)+ (4n2+4n 7).
n χ n − −
p
Proof Let S0 be a crown graphof order 2n with vertex set u ,u , ,u ,v ,v , ,v
n { 1 2 ··· n 1 2 ··· n}
and minimum dominating set =D = u ,v . Since χ(S0)=2, we have
{ 1 1} n
1 1 1 1 0 1 1 1
− ··· − − ···
 1 0 1 1 1 0 1 1 
− ··· − − ···
 ... ... ... ... ... ... ... ... ... ... 
 
 
 1 1 0 1 1 1 0 1 
 − − ··· − ··· 
 
 1 1 1 0 1 1 1 0 
A (S0)= − − ··· − ···  .
χ n  
 0 1 1 1 1 1 1 1 
 ··· − ··· − − 
 
 1 0 1 1 1 0 1 1 
 ··· − ··· − − 
 ... ... ... ... ... ... ... ... ... ... 
 
 
 1 1 0 1 1 1 0 1 
 ··· − − ··· − 
 
 1 1 1 0 1 1 1 0 
 ··· − − ··· − 2n 2n
  ×
Its characteristic polynomial is
λn 1[λ 1][λ 2]n 2[λ2+(2n 5)λ (6n 8)]
− −
− − − − −
26 P.SivaKotaReddy,K.N.PrakashaandGavirangaiahK
and its minimum dominating color eigenvalues are
(2n 5)+√(4n2+4n 7) (2n 5) √(4n2+4n 7)
0 1 2 − − − − − − −
specD(S0)= 2 2 .
χ n  
n 1 1 n 2 1 1
− −
 
The minimum dominating color energy of S0 is
n
(2n 5)+ (4n2+4n 7)
ED(S0) = 0(n 1)+ 1(n 1)+ 2 + − − −
χ n | | − | | − | | | 2 |
p
(2n 5) (4n2+4n 7)
+ − − − −
| 2 |
p
= (2n 3)+ 4n2+4n 7,
− −
p
i.e.,
ED(S0)=(2n 3)+ 4n2+4n 7.
χ n − − 2
p
.
Theorem 5.4 If K is a star graph of order n, then
1,n 1
−
(i) E (K )=√5 for n=2;
χ 1,n 1
−
(ii) E (K )=(n 2)+ (n2 2n+3)forn 3..
χ 1,n 1
− − − ≥
p
Proof LetK be acoloredgraphonn vertices. Minimumdominatingsetis D = v .
1,n 1 0
− { }
Then we have
1 1 1 1 1
···
 1 0 1 1 1 
− ··· − −
 1 1 0 1 1 
A (K ) − ··· − −  .
χ 1,n−1  ... ... ... ... ... ... 
 
 
 1 1 1 0 1 
 − − ··· − 
 
 1 1 1 1 0 
 − − ··· − n n
  ×
Case 1. The characteristicequationfor n=2 is λ2 λ 1=0 andthe minimum dominating
− −
1 √5
color eigenvalues for n=2 are = ± . Whence, ED(K )=√5.
2 χ 1,n−1
Case 2. The characteristicequationfor n 3 is (λ 1)n 2(λ2+(n 3)λ (2n 3))=0 and
−
≥ − − − −
The minimum dominating color eigenvalues for n 3 are
≥
(n 3)+√(n2+2n 3) (n 3) √(n2+2n 3)
1 − − − − −
2 2 .
 
n 2 1 1
−
 
MinimumDominatingColorEnergyofaGraph 27
Its minimum dominating color energy is
n 3+ (n2+2n 3)
ED(K ) = 1(n 2)+ − −
χ 1,n−1 | | − | p 2 |
n 3 (n2+2n 3)
+ − − −
| 2 |
p
= (n 2)+ (n2 2n+3).
− −
p
Therefore,
ED(K )=(n 2)+ (n2 2n+3).
χ 1,n−1 − − 2
p
Definition 5.5 The cocktail party graph, denoted by K , is graph having vertex set V =
n 2
×
n u ,v and edge set E = u u ,v v ,u v ,v u :1 i<j n . This graph is also called
i=1{ i i} { i j i j i j i j ≤ ≤ }
as complete n-partite graph.
S
Theorem 5.6 If K is a cocktail party graph of order 2n, then ED(K ) = (4n 5)+
n×2 χ n×2 −
(4n2 4n+9).
−
p
Proof LetK be a cocktailpartygraphoforder2nwithV(K )= v ,v , ,v ,u ,
n 2 n 2 1 2 n 1
× × { ···
u , ,u . The minimum dominating set =D = u ,v . Then,
2 n 1 1
··· } { }
1 1 1 1 1 1 1 1
− ···
 1 1 1 1 1 1 1 1 
− ···
 1 1 0 1 1 1 1 1 
 − ··· 
 
 1 1 1 0 1 1 1 1 
 − ··· 
ADχ(Kn×2)= ... ... ... ... ... ... ... ... ...  .
 
 1 1 1 1 0 1 1 1 
 ··· − 
 
 1 1 1 1 1 0 1 1 
 ··· − 
 
 1 1 1 1 1 1 0 1 
 ··· − 
 
 1 1 1 1 1 1 1 0 
 ··· − 2n 2n
  ×
Its characteristic equation is
[λ+3]n 2[λ 1]n 1[λ 2][λ2 (2n 5)λ 4(n 1)]
− −
− − − − − −
with the minimum dominating color eigenvalues
2n 5+√(4n2 4n+9) 2n 5 √(4n2 4n+9)
3 1 2 − − − − −
EspecD(K 2)= − 2 2 .
χ n×  
n 2 n 1 1 1 1
− −
 
28 P.SivaKotaReddy,K.N.PrakashaandGavirangaiahK
and the minimum dominating color energy,
2n 5+ (4n2 4n+9)
ED(K 2) = 3(n 2)+1(n 1)+ 2 + − −
χ n× |− | − − | | | 2 |
p
2n 5 (4n2 4n+9)
+ − − −
| 2 |
p
= (4n 5)+ (4n2 4n+9).
− −
p
This completes the proof.
2
Definition 5.7 The friendship graph, denoted by F(n), is the graph obtained by taking n copies
3
of the cycle graph C with a vertex in common.
3
Theorem 5.8 If F(n) is a friendship graph, then ED(F(n))=(3n 2)+ (n2+6n+1).
3 χ 3 −
p
Proof Let F(n) be a friendship graph with V(F(n)) = v ,v ,v ,...,v . The minimum
3 3 { 0 1 2 n}
dominating set =D = v . Then,
3
{ }
0 1 1 1 1 0
− ··· −
 1 0 1 0 0 1 
··· −
 1 1 1 1 1 1 
 ··· 
AD(F(n))== 1 0 1 0 1 0  .
χ 3  − ··· − 
 ... ... ... ... ... ... ... 
 
 
 1 0 1 1 0 1 
 − − ··· 
 
 0 1 1 0 1 0 
 − ··· (2n+1) (2n+1)
  ×
Its characteristic equation is
λn 1[λ+n][λ 2]n 1[λ2+(n 3)λ+(2 3n)]
− −
− − −
with the minimum dominating color eigenvalues
(n 3)+√(n2+6n+1) (n 3) √(n2+6n+1)
EspecD(F(n))= −n 0 2 − − 2 − − − 2 .
χ 3  
1 n 1 n 1 1 1
− −
 
and the minimum dominating color energy
(n 3)+ (n2+6n+1)
ED(F(n)) = n +0+ 2(n 1)+ − −
χ 3 |− | | | − | 2 |
p
(n 3) (n2+6n+1)
+ − − −
| 2 |
p
= (3n 2)+ (n2+6n+1).
−
p
This completes the proof.
2
MinimumDominatingColorEnergyofaGraph 29
§6. Properties of Minimum Dominating Color Energy of a Graph
Theorem6.1 Let λI AD =a λn+a λn 1+a λn 2+....+a bethecharacteristicpolynomial
| − C| 0 1 − 2 − n
of AD. Then
c
(i) a =1;
0
(ii) a = D ;
1
−| |
(iii) a =(D ) (m+m ).
2 2 c
| | −
where m is thenumberof edges and m is thenumberof pairs of non-adjacent vertices receiving
c
the same color in G.
Proof (i) It follows from the definition, P (G,λ):=det(λI A (G)), that a =1.
c c 0
−
(ii) The sumofdeterminants ofall1 1principalsubmatricesofAD is equalto the trace
× c
of AD, which a =( 1)1 trace of [AD(G)]= E .
c ⇒ 1 − c −| |
(iii) The sum of determinants of all the 2 2 principal submatrices of [AD] is
× c
a a
a = ( 1)2 ii ij = (a a a a )
2 − 1 i<j n(cid:12)(cid:12) aji ajj (cid:12)(cid:12) 1 i<j n ii jj − ij ji
≤X≤ (cid:12) (cid:12) ≤X≤
(cid:12) (cid:12)
= a a (cid:12) a(cid:12) a
ii jj (cid:12)− (cid:12)ji ij
1 i<j n 1 i<j n
≤X≤ ≤X≤
= (D 2) (m+number of pairs of non adjacent vertices
| − −
receiving the same color in G)
= (D 2) (m+m ).
c
| − 2
n n
Theorem 6.2 If λ ,λ , ,λ are eigenvalues of AD(G), then λ = D and λ2 =
1 2 ··· n c i | | i
i=1 i=1
D +2(m+mc), where mc is the number of pairs of non-adjacent Pvertices receiving tPhe same
| |
color in G.
§7. Open Problems
Problem 1. Determine the class of graphs whose minimum dominating color energy of a
graph is equal to number of vertices.
Problem 2. Determine the class of graphs whose minimum dominating color energy of a
graph equal to usual energy.
References
[1] V.Kaladevi and G.Sharmila Devi , Double dominating skew energy of a graph, Intern. J.
Fuzzy Mathematical Archive, Vol.4, No. 1, 2014, 1-7.
[2] M.R. Rajesh Kanna, B.N. Dharmendra and G. Sridhara, Minimum dominating energy of
30 P.SivaKotaReddy,K.N.PrakashaandGavirangaiahK
a graph, International Journal of Pure and Applied Mathematics, Volume 85 No. 4 2013,
707-718.
[3] I. Gutman,B. Zhou, Laplacian energy of a graph, Linear Algebra and its Applications,
414(2006),29-37.
[4] C.AdigaandC.S.Shivakumaraswamy,Boundsonthelargestminimumdegreeeigenvalues
of graphs, Int.Math.Forum., 5(37)(2010),1823-1831.
[5] C. Adiga, E. Sampathkumar and M. A. Sriraj, Color energy of a graph, Proc.Jangjeon
Math.Soc., 16(3)(2013), 335-351.
[6] R.Balakrishnan, Energy of a Graph, Proceedings of the KMA National Seminar on Graph
Theory and Fuzzy Mathematics, August (2003), 28-39.
[7] R. B. Bapat and S. Pati, Energy of a graph is never an odd integer, Bull. Kerala Math.
Assoc., 1(2004), 129-132.
[8] Fan. R.K.Chung,Spectral graph theory, CBMSRegionalConferenceSeriesinMathemat-
ics, No.92, 1997.
[9] C. A. Coulson, On the calculation of the energy in unsaturated hydrocarbons molecules.
Proc.Cambridge.Phil.Soc, 36(1940),201-203.
[10] D.M.Cvetkovi´c,M.Doob,HSachs,SpectraofGraphs-TheoryandApplications,Academic
Press, new York, 1980.
[11] I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz, 103(1978),
1-22.
[12] I. Gutman, Hyperenergetic molecular graphs, J. Serb.Chem., 64(1999),199-205.
[13] I. Gutman, The energy of a graph: old and new results, Combinatorics and Applications,
A. Betten, A. Khoner, R. Laue and A. Wassermann,eds., Springer, Berlin, 2001,196-211.
[14] V. Nikiforov,Graphs and matrices with maximal energy,J.Math. Anal. Appl., 327(2007),
735-738.
[15] S. Pirzada and I. Gutman, Energy of a graph is never the square root of an odd integer,
Applicable Analysis and Discrete Mathematics, 2(2008),118-121.
[16] H. S. Ramane, I. Gutman and D. S. Revankar, Distance equienergetic graphs, MATCH
Commun. Math. Comput. Chem., 60(2008), 473-484.
[17] H.S.Ramane,H.B.Walikar,S.B.Rao,B.D.Acharya,P.R.Hampiholi,S.R.Jog,I.Gut-
man, Spectra and energies of iterated line graphs of regular graphs,Applied Mathematical
Letters, 18(2005),679-682.
[18] E. Sampathkumar, Graphs and matrices, Proceeding’s of the Workshop on Graph Theory
Applied to Chemistry, Eds. B. D. Acharya, K. A. Germina and R. Natarajan, CMS Pala,
No 39, 2010, 23-26.
[19] E. Sampathkumar and M. A. Sriraj, Vertex labeled/colored graphs, matrices and signed
graphs, J. of Combinatorics, Information and System Sciences, to appear.
[20] D. Stevanovic and G. Indulal, The distance spectrum and energy of the composition of
regular graphs, Appl. Math. Lett., 22(2009), 1136-1140.
[21] T. Aleksic, Upper bounds for Laplacian energy of graphs. MATCH Commun. Math.
Comput. Chem., 60 (2008) 435-439.
[22] N. N. M. de Abreu, C. T. M. Vinagre, A. S. Bonifacio, I.Gutman, The Laplacian energy
MinimumDominatingColorEnergyofaGraph 31
of some Laplacian integral graphs. MATCH Commun. Math. Comput. Chem., 60(2008),
447 - 460.
[23] J. Liu, B. Liu, On relation between energy and Laplacian energy. MATCH Commun.
Math. Comput. Chem. 61(2009),403-406.
[24] G. H. Fath-Tabar, A. R. Ashrafi, I. Gutman, Note on Laplacian energy of graphs. Bull.
Acad. Serbe Sci. Arts (Cl. Math. Natur.), 137(2008),1-10.
[25] M.Robbiano,R.Jimenez,ApplicationsofatheorembyKyFaninthe theoryofLaplacian
energy of graphs. MATCH Commun. Math. Comput. Chem. 62(2009),537-552.
[26] D.Stevanovic,I.Stankovic,M.Milosevic,Moreonthe relationbetweenenergyandLapla-
cian energy of graphs. MATCH Commun. Math. Comput. Chem. 61(2009), 395-401.
[27] H.Wang,H.Hua,NoteonLaplacianenergyofgraphs. MATCHCommun. Math. Comput.
Chem., 59(2008), 373-380.
[28] B. Zhou, New upper bounds for Laplacian energy. MATCH Commun. Math. Comput.
Chem., 62(2009), 553-560.
[29] B.Zhou,I.Gutman,Nordhaus-Gaddum-typerelationsfortheenergyandLaplacianenergy
of graphs. Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.) 134(2007),1-11.
[30] B.Zhou,OnthesumofpowersoftheLaplacianeigenvaluesofgraphs. Lin. Algebra Appl.,
429(2008),2239-2246.
[31] B.Zhou,I.Gutman,T.Aleksic,AnoteonLaplacianenergyofgraphs. MATCH Commun.
Math. Comput. Chem., 60(2008),441-446.
[32] B. Zhou, I. Gutman, On Laplacian energy of graphs. MATCH Commun. Math. Comput.
[33] H. B. Walikar and H. S. Ramane, Energy of some cluster graphs, Kragujevac Journal of
Science, 23(2001),51-62.
[34] B.J. McClelland, Properties of the latent roots of a matrix: The estimation of pi-electron
energies. J. Chem. Phys., 54(1971), 640 - 643.
[35] M. Fiedler, Additive compound matrices and an inequality for eigenvalues of symmetric
stochastic matrices, Czech. Math. J., 24(99) 392-402(1974).