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Bounds for the Eigen Values and Energy of Degree Product Adjacency Matrix of A Graph

Keerthi G. Mirajkar* and Bhagyashri R. Doddamani1

*,1Department of Mathematics, Karnatak Arts College, Karnatak University

Dharwad - 580001, Karnataka, INDIA.

email:*keerthi.mirajkar@gmail.com, 1bhagyadoddamani1@gmail.com.

(Received on: March 14, 2019) ABSTRACT

In this article, we obtain a bound for eigen values of degree product adjacency matrix [DPA(G)] and also obtain some lower bounds for the degree product adjacency energy of graph G.

Mathematical Subject Classification:05C07, 05C50.

Keywords: Energy, Degree product adjacency energy.

1. INTRODUCTION

Let G be a simple, connected and finite graph with order n and size m. di is the degree of the vertex vi, where di is the number of edges incident to the vertex vi. For undefined terminologies we refer7. A molecular graph is a graph in which the vertices corresponds to the atoms and edges corresponds to the bonds. Chemical graph theory is a branch of mathematical chemistry which has an imporatant effect on the development of the chemical sciences.

The adjacency matrix A(G) of a graph G will be (0, 1) matrix and is defined as

 

 0 otherwise.

,

~ if

1

i j

ij

v a v

If 1, 2, 3, …, n are the eigen values of A(G). Then the energy of a graph G is defined as the sum of absolute values of the eigen values of adjacent matrix of graph G. This concept was introduced by I. Gutman in5.

(2)

n

i i

A G

E

1

)

(

The above defined energy explained is the way of pure mathematical concept. But when a molecular graph is used to model a -electron system, the energy of the graph has been shown to be a good approximation of the binding energy of the -electrons. Motivation of the above defined energy has been well explained in6. The total molecular orbital energy of all - electrons in a molecule, by Hückel molecule orbit model is given by

n

i i i

ne

E

1

Where ne corresponds to the energy of ne isolated p-electron, is a constant, i is the number of -electrons in the ith molecular orbital and i's are the eigen values of the corresponding molecular graph. Since the author in5 was interested only in the binding energy of the - electrons, so he considered the non-trivial part of the above equation i.e.,

n

i i

E i 1

. For most conjugated -electron systems of chemical interests, all bonding molecular orbitals are doubly occupied and antibonding molecular orbitals are unoccupied. This leads to the fact that

 

 

. 0 if , 0

, 0 if , 2

i i

i

Hence

2 0

i i

E . Notice that for a simple graph

( ( ))0

G A trace

n

n i

i then .

1

n

i

E

i The Cauchy-Schwarz inequality2 states that if (a1, a2, a3, …, an) and (b1, b2, b3, …, bn) are real n-vectors then,

 

 

 

 

 

   

n

i i n

i i n

i i

i

b a b

a

1 2

1 2 2

1

.

Motivated by the work of C. Adiga et al.1 and H. S. Ramane et al.11, we introduce the concept of degree product adjacency energy8, which is defined as follows.

Definition

Let G be a simple, connected graph with n-vertices v1,v2,v3, …, vn and di be the degree of the vertex vi, ∀i = 1, 2, …, n. Then the degree product adjacency matrix [DPA(G)] of a graph G is [dij] i.e.,

 

 0 , otherwise.

,

~ if

,

i j

j i ij

v v d

d d

The degree product adjacency matrix [DPA(G)] is a real symmetric matrix and its eigen values are 1, 2, 3, …, n. The order of eigen values be arranged as 1 2 3 ≥ …≥ n. In the

(3)

similar way of adjacency energy, the degree product adjacency energy of a graph defined and is denoted by as follows,

. )

(

1

n

i i

DPA G

E

In this article, we establish the results on bound for the largest eigen value of DPA(G) and also obtain the lower bounds for the degree product adjacency energy of a graph G.

2. RESULTS

To present the complete results, some important theorems which are used throught out the paper are mentioned below.

Theorem 2.1.10 Suppose ai and bi, 1≤ i ≤ n are positive real numbers, then

2

1 2

2 1

2 1

2 1

2 1

1 2

1 2

4

1 

 

 

 

 

 

n

i i i n

i i n

i

i

a b

M M

m m m

m M b M

a

Where

M

1

 max

1in

( a

i

); M

2

 max

1in

( b

i

); m

1

 min

1in

( a

i

); m

2

 min

1in

( b

i

)

Theorem 2.2.9 Let ai and bi, 1≤ i ≤ n are nonnegative real numbers, then

1 2 1 2

2

2 2

1 1

2

1 2

4 M M m m

b n a b

a

n

i i i n

i i n

i

i

  

 

  

where M1M2 and m1m2 are defined similarly to Theorem 2.1.

Theorem 2.3.3 Suppose ai and bi, 1≤ i ≤ n are positive real numbers, then

)

)(

)(

(

1 1 1

b B a A n b

a b

a n

n

i i n

i i n

i i

i

     

where a, b, A and B are real constants, that for each i, 1≤ i ≤ n, a≤ai≤ and b≤bi ≤ Further,

 

 

 

 

 

 

 

 

2 1 1 ) 2

( n

n n n

n

.

Theorem 2.4.4 Let ai and bi, 1≤ i ≤ n are nonnegative real numbers, then

 

 

  

n

i i i n

i i n

i

i

rR a r R a b

b

1 1

2

1

2

( )

where r and R are real constants. So that for each i, 1≤ i ≤ n holds raibi≤ ai

2.1 Bounds for the largest eigen value of DPA(G)

We need the following Lemma to prove the further results.

(4)

Lemma A. If the trace [DPA(G)] = 0, then the eigen values obtained from DPA(G) matrix satisfies the following,

 

P

d d

d

j i d d

G DPA trace

n j i n

i n

j ij n

i n

j ji ij n

i i n

i i

2

) ( 2

~ ,

)) ( ( )

ii (

0 (i)

j i 1

2

1 1

2

1 1

1 2 2 1

 

 

Where

n j

i

d

d P

j i 1

)

2

(

Theorem 2.5. If G be a graph with n-vertices, then

n n P ( 1 ) 2

1

 

Proof. Consider the graph G with n-vertices. Let DPA(G) be the degree product adjacency matrix of graph G and 1, 2, 3, …, n are the eigen values obtained from the DPA(G) matrix, where 1 is the largest eigen value and the bound for 1 is calculated by using cauchy-schwarz inequality i.e.,

 



 



 

  

n

i i n

i i n

i i

ib a b

a

1 2 1

2 2

1

.

Let ai =1 and bi = i , ∀i = 2, 3, …, n then the inequality becomes,

 



 



 

  

n

i i n

i n

i i

1 2 1

2 2

1

. 1 )

)(

1

(

(1) From Lemma A(i),

2 1 2

2 2 1 1

) (

0 0

 

 

n

i i n

i i n

i i

(2)

(5)

And from Lemma A(ii)

 

   

 

1 2

2 2 2

2 2

1 1

2

) ( 2

2 2

P P P

n

i i n

i i n

i i

(3)

Substituting (2) and (3) in (1), we get

 

n n P

n n

P P n

i i i

) 1 ( 2

) 1 ( ) 1 ( 2

) 2

)(

1 (

1

2 2

1

2 2

 

The equality relation for 1 holds for the regular graphs, P2 and W4. Theorem 2.6. If G be a graph with n-vertices, then

√2 ≤ ( ) ≤ √2

Proof. Consider a graph G with n-vertices. Let DPA(G) be the degree product adjacency matrix of a graph G and 1, 2, 3, …, n are the eigen values of the matrix DPA(G). Now we consider the cauchy-schwarz inequality to prove the theorem,

 Proof for Right hand side bond:

Let us assume that ai = 1 and bi =| i|, ∀ = 1, 2, …, n.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

1 2

1

1 2

1 2 2

1

1 2

1 2 2

1

|

|

|

|

|

| . ) 1 (

|

| ) 1 (

.

n

i i n

i i

n

i i n

i n

i

i

n

i i n

i i n

i i i

n

b a

b a

After simplification by using the Lemma A.

( ) ≤ √2 (4)

(6)

 Proof for Left hand side bond:

We know that,

 

n

i i n

i i

1 2 2

1

|

|

By using the Lemma A, we conclude that

( ) ≥ √2 (5) From equation (4) and (5),

√2 ≤ ( ) ≤ √2

2.2 Lower bounds for the degree product adjacency energy [EDPA(G)]

Theorem 2.7. Let G be a graph with n-vertices and m-edges. Suppose | 1|≥| 2|≥| 3|≥ …≥| n| are the eigen values of DPA(G), then the following inequality holds.

|

|

|

|

|

||

| 2 ) 2 (

1 1

n n DPA

G nP

E

 

Proof. Consider a graph G with n-vertices and | 1|≥| 2|≥| 3|≥ …≥| n| are the eigen values of DPA(G), where | 1| and | n| are the maximum and minimum eigen values of | i|'s respectively.

We have the inequality by the theorem 2.1,

2

1 2

2 1

2 1

2 1

2 1

1 2

1 2

4

1 

 

 

 

 

 

n

i i i n

i i n

i

i

a b

M M

m m m

m M b M

a

Assume ai =1, bi=| i|, M1M2=| 1| and m1m 2=| n| then,

2

1 2

1 1

1 2

1

2

( 1 )

4

1 1 

 

 

n

i

i n

n n

i i n

i

From Lemma A,

 

 

 

 

n n DPA

n n DPA

DPA n

n

G nP E

G nP E

G E P

n

 

 

1 1

2 1

1 2

2

1 2 1

2 ) 2 (

) 8 (

) 4 (

2 1

(7)

Theorem 2.8. Let G be a graph with n-vertices, then the following inequality holds.

1

2

2

|

|

| 4 | 2

)

(

n

DPA

nP n G

E  

Proof. Consider a graph G with order n and size m. Let | 1|≥| 2|≥| 3|≥ …≥| n| be the eigen values of DPA(G) matrix, where | 1| and | n| are the maximum and minimum eigen values respectively.

From Theorem 2.2 we have the inequality,

1 2 1 2

2

2 2

1 1

2

1 2

4 M M m m

b n a b

a

n

i i i n

i i n

i

i

  

 

  

Assume ai = 1, bi = | i|, M1M2 =| 1| and m1m2 = | n| in the above inequality,

1

2

2 2

1 1

2

1 2

) 4 1 (

1

n

n

i

i n

i i n

i

n

  

 

  

From Lemma A,

 

1

2

2 2 1

2 2

2 4 ) (

)) 4 ( ( 2

n DPA

n DPA

nP n G

E G n E P n

Theorem 2.9. Let G be a graph with n-vertices, then the following inequality holds.

1

2

) ( 2

)

(

n

DPA

G nP n

E  

Proof. Consider a graph G with order n and size m. Let | 1|≥| 2|≥| 3|≥ …≥| n| be the eigen values of DPA(G) matrix, where | 1| and | n| are the maximum and minimum eigen values respectively.

Consider inequality from the Theorem 2.3,

) )(

)(

(

1 1 1

b B a A n b

a b

a n

n

i i n

i i n

i i

i

     

Now assume that ai = bi = | i|, A = B = | 1| and a = b = | n|, then the inequality reduces to

)

)(

)(

(

1 1

2

1 1

2

n n

n

i i n

i

i

n

n  

 

  

From Lemma A,

 

2 1

2 1

2

) )(

( 2

) (

) )(

( )

( 2

n DPA

n DPA

n nP G

E

n G

E nP

(8)

Theorem 2.10. Let G be a graph with n-vertices, then the following inequality holds.

n n DPA

n G P

E

 

1

2

1

) (

Proof. Consider a graph G with order n and size m. Let | 1|≥| 2|≥| 3|≥ …≥| n| be the eigen values of DPA(G) matrix, arranged in non-increasing order, where | 1| and | n| are the maximum and minimum eigen values respectively.

We make use of the inequality from the Theorem 2.4,

 

 

  

n

i i i n

i i n

i

i

rR a r R a b

b

1 1

2

1

2

( )

Assume bi = | i|, ai = 1, r = | n| and R = | 1|, then the inequality implies to

 

 

 

  

n

i

i n

n

i n

i

n i

1 1 1

2

1

1

2

1 ( 1 )

From Lemma A,

   

1 1 1 1

) ( ) 2

(

) ( )

( 2

 

n n DPA

DPA n

n

n G P

E

G E n

P

ACKNOWLEDGEMENT

The authors are thankful to university Grants Commission (UGC), Govt. of India for financial support through research grant under F1-17.1/2017-18/RGNF-2017-18-SC-KAR- 39176 /(SA-III/Website).

REFERENCES

1. C. Adiga and M. Smitha, On maximum degree energy of a graph, International Journal of Contemporary Mathematical Sciences, 4(8), 385-396 (2009).

2. S. Bernard and J. M. Child, Higer Algebra, Macmillan India Ltd., New Delhi, (2001).

3. M. Biernacki, H. Pidek and C. Ryll-Nardzewsk, Sur une iné galité entre des intégrales définies, Maria Curie SkÅĆodowska University, A4, 1-4 (2009).

4. J. B. Diaz and F. T. Metcalf, Stronger forms of a class of inequalities of G. Pólya-G. Szegȍ and L. V. Kantorovich, Bulletin of the AMS-American Mathematical Society, 69, 415-418 (1963).

5. I. Gutman, The energy of a graph, Berlin Mathematics-Statistics Forschungszentrum, 103, 1-22 (1978).

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6. I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer- verlag, Berlin, (1986).

7. F. Harary, Graph Theory, Addison-Wesley, Reading, (1969).

8. K. G. Mirajkar and B. R. Doddamani, On energy and spectrum of degree product adjacency matrix for some class of graphs. (Communicated).

9. N. Ozeki, On the estimation of inequalities by maximum and minimum values, Journal of College Arts and Science, Chiba University, 5, 199-203 (1968).

10. G. Pólya and G. Szegȍ, Problems and theorems in analysis, Series, Integral calculus, Theory of functions, Springer, Berlin, (1972).

11. H. S. Ramane, D. S. Revankar and J. B. Patil, Bounds for the degree sum eigen values and degree sum energy of a graph, International Journal of Pure and Applied Mathematical Sciences, 6(2), 161-167 (2013).

References

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