Inverse Sum Status Energy of a Graph

9

VI

https://doi.org/10.22214/ijraset.2021.34849

June 2021

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 9 Issue VI Jun 2021- Available at www.ijraset.com

Inverse Sum Status Energy of a Graph K. N. Prakasha1, Padmanava Samanta2, Niyati Misra3 1

Department of Mathematics, Vidyavardhaka College Engineering, Mysuru-570002, INDIA 2 Department of Mathematics, Berhampur University, Odisha, INDIA

Abstract: Motivated by the inverse sum status index, we introduce the inverse sum status matrix ISS (G ) as ISS (G )  ISS nn = (

+

~ ,

)

Thus we also obtained the results for well known graphs.

Keywords: Inverse sum status energy, Inverse Sum Indeg matrix. 2010 AMS Subject Classification: 05C50.

I. INTRODUCTION Status of the vertex ‘u’ is defined as the sum of distance of all other vertices from ‘u’… Inverse Sum Status index is introduced by V. R. Kulli [35]. He defined it as

ISS G  

 u v  u   v 

Where  u represents the status of the vertex u.

Motivated by the Inverse Sum Status index, We introduce the Inverse Sum Status matrix

ISS (G ) as ISS (G )  ISS nn

 u v      if vi ~ v j v  u ISS   0 otherwise   The Inverse Sum Status energy is given by n

ISSE(G )    i

(1)

i 1

The inverse sum status energy is given by

Figure 1. Graph with 6 vertices ( )=

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International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 9 Issue VI Jun 2021- Available at www.ijraset.com II.

INVERSE SUM STATUS ENERGY OF SPECIFIC GRAPH STRUCTURES

1) Theorem-2.1. Inverse sum status energy of complete graph

Kn

is

ISSE K n   ISIE K n   2 2 n  4 . [?]

K n ,  u   n  1 . Then the every ijth – entry of the Inverse Sum Status matrix will be

Proof. For each and every vertex u in n – 1.

 0 n  1  n  1 ISS K n      n  1  n  1

n 1

n 1



n 1

0 n 1  n 1 n 1

n 1 0   

   n 1 n 1

n 1 n 1  0 n 1

n  1 n  1  n  1    n  1  0 

This matrix will be same as Inverse Sum Indeg Matrix. Then the energy will be also as same as Inverse Sum indeg energy.

Therefore,

2

ISSE K n   n  1

.

2) Theorem-2.2. The inverse sum status energy of the star

K1,n1

n 1 , 2 n1

ISSEK1,n1  

Proof. The status of the central vertex in n – 1 and the pendant vertices is 2n – 3. Thus the inverse sum status matrix is     2n   2n K 1 , n  1      2n   2n  

2n

0 2

 3n 2  3n

5n   4 5n   4  2  5n  3n  4 2  5n  3n  4

2

 5n  3 2n 2  5n  3 & & ...

3n  4 3n  4

3 3

3 3

Clearly, the characteristic equation is

 2n2  3n 2n2   3n  2n2   3n 2n2   3n 



2n 2  5n  3 2n 2  5n  3 & 3n  4 3n  4

0

0



0

0

0



0









0

0



0

0

0



0

   0   0    0  0 

 I  ISS K1,n1   0

2n 2  5n  3 2n 2  5n  3 &  & ...

3n  4 3n  4

5n  3  4 5n  3  4 5n  3  4 5n  3  4

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

2 n 2  5n  3 2n 2  5n  3 &  3n  4 3n  4



0



0

0

0





0









0  0 

0

0





0

0

0



0



87

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 9 Issue VI Jun 2021- Available at www.ijraset.com



 n 12n 5n  3    n 12n 5n  3  0   2

n2 

 

2

 

3n  4



 

3n  4



 

. Therefore, the spectrum would be





 n 1 2n2  5n  3  n 1 2n2  5n  3    0 SpecISSK1,n1    3n  4 3n  4   1 1 n  1   Therefore,

2 n12n 5n3 ISSEK   3n  4 2

1,n1

3) Theorem-2.3. The inverse sum status energy of crown graph

S n0

.

is

 

ISSE S n0  6nn  1 .

Proof. The inverse sum status matrix is

 0  0  0     0  0   3n  2    3n 2  3n  2

0

0



0

0

0



0

0

0



0









0

0



0

3n 2 3n 2  3n 2 3n 2

3n 3n  2 2 3n 0  2    3n 0  2 3n  0 2

3n 2 3n 2  3n 2

3n 3n  2 2 3n 0  2 3n  0 2    3n 3n  2 2

0

0



0

0

0



0









0

0



0

0

0



0

0

3n  2 3n   2 3n  2    0  0   0    0   0 

 I  RTZ 3 K n  

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     0   0      0    0  3n   2     3n  2  3n   2

0

0



0



0



0

0





0









0

0



0



3n 2 0

 3n  2 3n  2

0 3n 2 3n  2  3n  2 

3n 3n   2 2 3n 3n    2 2    3n 0   2 3n   0 2 

3n 3n   2 2 3n 0   2 3n   0 2    3n 3n    2 2 



0



0

0





0









0

0





0

0



0

3n  2 3n    2 3n   2    0  0  0   0     0       

Characteristic equation is

3n      2 

n 1

3n      3  

n 1

3n  3n     n  1     n  1    0 2  2  

  0

spectrum is Spec ISS S n

3n 3n 3n   3n  n  1  n  1     2 2 2 2   1 1 n  1 n  1 

 

Therefore, ISSE S n  6 nn  1 .

0

4) Theorem-2.4. For complete bipartite graph, the Inverse Sum Status energy is 5

ISSE K m , n   2mn  2 .









2

 0 mm  J nm

Proof. ISS K m , n  ISS K m ,n  mn  

 0 Spec ISS K m ,n    m  n  2 





5

mn 2 1

J mn   0 nn  5  mn  2  1 

5

Therefore, ISSE K m , n  2mn  2 .

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International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 9 Issue VI Jun 2021- Available at www.ijraset.com 5) Theorem-2.5. The Inverse Sum Status energy of K n2 is

ISSE  K n2   2nn  1 .

Proof. Here the status of each vertex is 2n. hence Inverse Sum Status matrix is 0 0  n  n   n n  n n 

Spec

ISS

0 0

n n

n n

 

n n

n n

n n

n n  n n

0 0  n n

0 0  n n

    

n n  0 0

n n  0 0

n n  n n

n n

n n

n n

 

n n

n n

0 0

K n 2  

 n n  1 0  1 n 

n n  n  n   n n  0 0 

n   n  1 

ISSE K n2   2nn  1 .

III. PROPERTIES OF INVERSE SUM STATUS ENERGY The following results will give the initial coefficients of the Inverse Sum Status characteristic polynomial.

1) Proposition-3.1. In the Inverse Sum Status characteristic polynomial

 ISS G,   , the first three coefficients are 1,0 and

2

n

  u v     respectively. i 1   u   v   2) Proposition-3.2. n

n

  u v    2   i 1 i 1   u   v  

2

2 i

(Where If

 i represents Inverse sum status eigenvalues.)

Proof. We know that n

 i 1

n

n

 i2   i 1

a

ij

a ji

j 1

n

 2 aij2   aii2 i j

i 1

 2 a ij2 i j

n

2

  u v   2   . i 1   u   v  

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International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 9 Issue VI Jun 2021- Available at www.ijraset.com 3) Theorem-1.3. Let G be a graph with n vertices. Then

ISSE G  

n

  u v  2n    i 1   u   v  

2

Proof. We have Cauchy-Schwartz inequality as 2

 n   n  n    xi y i     xi2    y i2   i 1   i 1   i 1  Substitute xi  1 and y i 

 i . Then 2

 n   n  n 2    i    1    i   i 1   i 1   i 1  Which implies that 2

   n   u v   2n  i 1       u   v   u v   

ISSE  G  

4) Theorem-1.4.

2

n

2   u v  n  n  n  1   Det  ISS  G        u   v  

ISSE (G )  2 i 1

Proof. By definition, 2

ISSEG 

 n     i   i 1  n



2

n



i

i 1



j

j 1

 n    i 1

2 i   



i

j .

i j

Using arithmetic mean and geometric mean inequality, we have 1

1 nn  1



i

j

i j

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  n n 1     i  j  . i  j  

91

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321-9653; IC Value: 45.98; SJ Impact Factor: 7.429 Volume 9 Issue VI Jun 2021- Available at www.ijraset.com Therefore, 1 n

ISSE G 2  

i

i 1

2

  nn 1  nn  1   i  j   i j  1

n



 i 1

 n 2  i  nn  1   i  i 1

2  n 1  n  n 1  

2  n  u v   2    nn  1 ISS n .  i 1  u   v  

2  n  u v  ISSE G   2    nn  1ISS n ...  i 1  u   v  

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

C.Adiga and C.S.Shivakumaraswamy, Bounds on the largest minimum degree eigenvalues of graphs, Int. Math. Forum, 5(37)(2010), 1823-1831.

I. Gutman, The energy of a graph: old and new results, Combinarotics and applications, A. Betten, A. Khoner, R. Laue and A. Wassermann, eds., Sp[ringer, Berlin, 2001, 196-211.

F. Harry, On the notion of balance of a signed graph, Michigan Math. J., 2(1953-54), 143-146 and addendum proceeding, p.1.

F. Harary, On local balance and N-balance in signed graphs, Michigan Math. J., 3(1955), 37-41.

Kinkar Ch. Das, Bounds on the greatest eigenvalue of a graphs, Indian J. Pure appl. Math., 334(6), June 2003, 917-925.

Mohsen Molahajiaghaei, The eigenvalues and energy of integral circulant graph, Transactions on Combinatorics, 1(2012), No.3, 47-56.

V. Nikiforov, Graphs and matrices with maximal energy, J. Math. Anal. Appl., 327(2007), 735-738.

H.S. Ramane, H.B. Walikar, S.B.Rao, B.D. Acharya, P.R. Hampiholi, S.R., Jog, I. Gutman, Spctra and energies of iterated line graphs of regular graphs, Applied Mathematical Letters, 18(2005), 679-682.

E.Sampathkumar, P.Siva Kota Reddy and M.S. Subramanya, (4,d)-sigraph and its applications, Advan. Stud. Contemp. Math., 20(1)(2010), 115-124.

E.Sampathkumar, P.Siva Kota Reddy and M.S. Subramanya, Directonally n-signed Graphs-II, International. J. Math. And Combin, 4,89-98, (2009).

E.Sampathkumar and M.A.Sriraj, Vertex Labeled / Colored Graphs, Matrices and Signed Graphs, J. of Combinatorics, Inforamtion and System Sciences, to appear.

D.Stevanovic and G.l Indulal, The distance spectrum and energy of the composition of regular graphs, Appl. Math. Lett., 22(2009), 1136-1140.

H.B.Walikar, H.S.Ramane, and P.R.Hampiholi, On the energy of graph, Proc. Of the Conference on Graph connections (R. Balakrishnan, H.M.Mulder and A. Vijaykumar, ed.) Allied Pub. Ltd., New Delhi, 1999, 120-123.

T. Zaslavsky, Signed Graphs, Discrete Appl. Math., 4(1)(1982), 47-74.

T.Zaslavsky, A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas, VII Edition, Electronic J. Combinatorics, 8(1)(1998), Dynamic Surveys # 8, 124 pp. (Also, see VIII Preliminary Edition, March 2008).

T.Zaslavsky, Matrices in the Theory of Signed Simple Graphs, RMS-Lecture Notes Series 13, (2010), 207-229.

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