Schultz Energy of Some Graph

INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 8, ISSUE 11, NOVEMBER 2019 ISSN 2277-8616

Schultz Energy Of Some Graph

M. R. Rajesh Kanna, R. Jagadeesh , H. L. Parashivamurthy

Abstract : In this article we defined Schultz energy [1]. By using of the adjacency degree and distance matrices. This article contains computation of Schultz energies for some standard graphs like star graph, complete graph, crown graph, cocktail graph, complete bipartite graph and friendship graphs. At the end of this article upper and lower bounds for Schultz energy are also presented.

Index Terms: Schultz matrix, Schultz eigenvalues, Schultz energy. ————————————————————

1 INTRODUCTION 1.2 Schultz energy Study on energy of graphs goes back to the year 1978, when Let G be a simple graph of order n with vertex set V and edge I. Gutman [12] defined this while working with energies of set E(G). Let d(v )denotes the degree of the vertexv . Schultz conjugated hydrocarbon containing carbon atoms. All graphs matrix of a graph G is n × nmatrix and is defined by S(G) ≔ considered in this paper are assumed to be simple without (s ), where loops and multiple edges. Let A = (a ) be the adjacency matrix s = [d(v ) + d(v )] d(v , v ). of the graph G with its eigenvalues λ , λ , λ , … , λ assumed in The characteristic polynomial of S(G) is denoted by decreasing order. Since A is real symmetric, the eigenvalues f (G, ρ) = det (ρI − S(G)). of G are real numbers whose sum equal to zero. The sum of The Schultz eigenvalues of the graph G are the eigenvalues the absolute eigenvalues values of G is called the energy E(G) ofS(G) . Since S(G) is real and symmetric, its eigenvalues are of G. real numbers and we label them in non-increasing order

ρ ≥ ρ ≥ ρ ≥ ⋯ ≥ ρ . The Schultz energy of G is defined as i. e. , ℇ(G) = ∑|λ |. ℇ (G) ≔ ∑|ρ | Theories on the mathematical concepts of graph energy can be seen in there views [13], papers [8, 9, 14] and the Here trace of S(G) = 0. references cited there in. For various upper and lower bounds for energy of a graph can be found in papers [8, 9] and it was 2 MAIN RESULTS AND DISCUSSION observed that graph energy has chemical applications in the molecular orbital theory of conjugated molecules [15, 16]. 2.1 Properties of Schultz eigenvalues Theorem 2.1 Let G be a graph with vertex set V = 1.1 Schultz index *v , v , … , v +, edge set E. If λ , λ , λ , … , λ are the eigenvalues In 1989, H.P. Schultz [23] introduced a new topological index, of Schultz matrix S(G) then namely Schultz index which is defined as

푆(퐺) = ∑ [푑(푣 ) + 푑(푣 )] 푑(푣 , 푣 ). (i) ∑ λ = 0 (ii) ∑ λ = ∑(d + d ) .

{ , } ( )

Proof i) We know that the sum of the eigenvalues of S(G) is the where d(v , v ) is the shortest distance between vertices v and trace of S(G). v , and d(v ) is thedegree of the vertex v in G, Similarly d(v ). A. Dobrynin and Amide A. Kochetova in 1994 [1] also ∴ ∑ λ = ∑ s = 0. proposed the above index and called it the degree distance of a graph. (ii) Similarly the sum of squares of the eigenvalues of S(G) is

trace of ,S(G)-

∴ ∑ λ = ∑ ∑ s s

______

= ∑(s ) + ∑ s s  Rajesh Kanna. M. R Department of Mathematics, Sri D D Urs

Government First Grade College, Hunsur, India, PH - 09448150508. E-mail: [email protected]

 Government Science College (Autonomous), Nrupathunga Road, = ∑(s ) + 2 ∑(s )

Bangalore - 560 001, India PH-9110296180. E-mail: [email protected]

 Parashivamurthy H. L. S Research scholar, department of = ∑(s ) + 2 ∑([d(v ) + d(v )]d(v , v )) Mathematics, Bharathiar University, Coimbatore- 641046, India /

Department of Mathematics, BGS Institute of Technology,

Adichunchanagiri University, B.G. Nagar-571448, Nagamangala, = 2 ∑(d + d ) d Mandya District, India. PH - 9900262656. Email: [email protected]

= 2M, where = ∑(d + d ) d

2.2 Bounds for Schultz energy a graph, an upper bound for ℇ (G) is given in the following MClelland's [20] gave upper and lower bounds for ordinary theorem. energy of a graph. Similar bounds for ℇ (G) are given in the following theorem. ∑ ( ) Theorem 2.4 If G is a (n, m) graph with ( ) ≥ n

Theorem 2.2 Let G be a simple graph with n vertices and m Then edges and P = |detS(G)| then ℇ (G)

2 ∑ (d + d ) d √2 ∑(d + d ) d + n(n − 1)P ≤ ℇ (G) ≤ ( ) n

2 ∑ (d + d ) d 2 ∑ (d + d ) d + √(n − 1) [ − ( )] ≤ √n (2 ∑(d + d ) d ) . n n

Proof Proof Cauchy Schwarz inequality is Cauchy-schwartz inequality is

[∑ a ] ≤ (∑ a ) (∑ b ) (∑ a b ) ≤ (∑ a ) (∑ b )

Put

,ℇ (G)- ≤ n (2 ∑(d + d ) d ) . , ( ) - ( ) ℇ G − λ ≤ n − 1 (2 ∑(d + d ) d ) − λ [From theorem 2.1]

ℇ (G) ≤ λ + √(n − 1) (2 ∑(d + d ) d ) − λ ⇛ ℇ (G) ≤ √n (2 ∑(d + d ) d ) .

Since arithmetic mean is greater than or equal to geometric mean we have Let f(x) = x + √(n − 1) (2 ∑(d + d ) d ) − x ( ) 1 ∑|λ | |λ | ≥ *∏|λ ||λ |+ n(n − 1) For decreasing function f (x) ≤ 0 = x(n − 1) 1 − ≤ 0 ( )

( ) √(n − 1) .2 ∑ (d + d ) d / − x = [∏|λ | ] = [∏|λ |]

2 ∑ (d + d ) d = |∏ λ | = |detS(G)| = P ⇛ x ≥ √ n

Type equation here.

Now consider, ,ℇ (G)- ≤ (∑|λ |) 2 ∑ (d + d ) d 2 ∑ (d + d ) d since ( ) ≥ n, we have√ n n

= ∑|λ | + ∑|λ ||λ | 2 ∑ (d + d ) d ≤ ≤ λ n

√ ℇ (G) ≥ 2 ∑ (d + d ) d + n(n − 1)P . 2 ∑ (d + d ) d ∴ f(λ ) ≤ f ( ) n Theorem 2.3 If λ (G) is the largest minimum Schultz ( ) 2 ∑ (d + d ) d eigenvalues of S(G) then λ (G) ≥ . i. e. , ℇ (G) ≤ f(λ ) ≤ f ( ) n Proof For any non zero vector X, we have by[3], λ (A) =

2 ∑ (d + d ) d max * } i. e. , ℇ (G) ≤ f ( ) n J AJ 2 ∑ (d + d )d ∴ λ (G) = = J J n S(G) = where J is unit column matrix. n Just like Koolen and Moulton’s [19] upper bound for energy of

i. e. , ℇ (G)

∑ b + rR ∑ a ≤ (r + R) ∑ a b 2 ∑ (d + d ) d ≤ n Put b = |λ |, a = 1, r = |λ | and R = |λ | then

2 ∑ (d + d ) d + √(n − 1) *2 ∑(d + d ) d − ( ) + ∑|λ | |λ ||λ | ∑ 1 ≤ ( |λ | + |λ |) ∑|λ | n

i. e. , 2 ∑(d + d ) d + |λ ||λ |n ≤ (|λ | + |λ |)ℇ (G) Milovanovi’c [21] bounds for minimum Schultz energy of a graph are given in the following theorem. 2 ∑ (d + d ) d + n|λ ||λ | ∴ ℇ (G) ≥ (|λ | + |λ |) Theorem 2.5.Let G be a graph with n vertices and m edges. The question of when does the graph energy becomes a Let |λ | ≥ |λ | ≥ ⋯ ≥ |λ | be a non-increasing order of Schultz rational number was answered by Bapat and S. Pati in their eighenvalue of S(G) then paper [4]. similar result for Schultz energy is obtained in the following theorem. ℇ (G) ≥ √n .∑ (d + d ) d / − α(n)(|λ | − |λ |)

n 1 n Theorem 2.7.If the minimum Schultz energy ℇ (G) is a rational where ∝ (n) = n 0 1 (1 − 0 1) 2 n 2 number then ℇ (G) = |C|(mod2). and ,x-denotes the integral part of the real number Proof is similar to theorem 3.7 of [2] Proof For real numbers a,a , a , … , a , A and b, b , b , … , b ,B with a≤ a ≤ A and b ≤ b ≤ ∀ i = 1,2. . n the following 2.3 Schultz energy of some standard graphs inequality is valid. Theorem 2.8 For n≥ 2, the Schultz energy of some complete |n ∑ a b − ∑ a ∑ b | ≤∝ (n)(A − a)(B − b)where ∝ (n) graph K is 4(n − 1) .

n 1 n = n 0 1 (1 − 0 1) Proof Let K be a complete graph with vertex set V 2 n 2 ={v1,v2,v3,…..,vn}. Then the Schultz matrix of complete graph and equality holds if and only if a = a = ⋯ = a and b = is, b =. . . = b .

If a = |λ |, b = |λ |, a = b = |λ | and A=B=|λ |, then

|n ∑|λ | − (∑|λ |) | ≤ ∝ (n)(|λ | − |λ |)

But

Characteristics polynomial is

∑|λ | = 2 ∑(d + d ) d and ℇ (G) (−1) ,λ − 2(n − 1) -,λ + 2(n − 1)- . Characteristics equation is (−1) ,λ − 2(n − 1) -,λ + 2(n − 1)- = 0. 2(n − 1) −2(n − 1) ≥ √n (2 ∑(d + d ) d ) Spec(K ) = . / 1 n − 1 The Schultz energy is, then the above inequality becomes ℇ (K ) = |2(n − 1) |(1) + |−2(n − 1)|(n − 1)

ℇ (K ) = 2(n − 1) + 2(n − 1) ℇ (K ) = 4(n − 1) . n (2 ∑(d + d ) d ) − (ℇ (G)) ≤ ∝ (n)(|λ | − |λ |)

Theorem 2.9.The Schultz energy of star graph K , is 4 if n = 2

{ i. e. , ℇ (G) ≥ √n (2 ∑(d + d ) d ) −∝ (n)(|λ | − |λ |) 4(n − 2) + √n + 3n − 16n + 16 if n > 2

Theorem 2.6 Let G be a graph with n vertices and m edges. Proof Let K , be a star graph with vertex set V={v0,v1,v2,..,vn-1}. Let |λ | ≥ |λ | ≥ ⋯ ≥ |λ | > 0 be a non- increasing order of eigenvalues of S(G) then Case1; if n>2 Then the Schultz matrix of star graph is, 2 ∑ (d + d ) d + n|λ ||λ | ℇ (G) ≥ (|λ | + ||λ ||)

Proof Let 푎 ≠ 0, b r and R be a real numbers satisfying ra ≤ b ≤ Ra ,then the following inequality holds.[Theorem 2,[21]]

0 8(n − 1) −8(n − 1) 0 n n n  n S(K ) = . /   × n − 1 1 n n 0 4 4  4 The Schultz energy is, ℇ (K ) = |0|(n − 1) + |8n(n − 1)|(1) + |−(n − 1)|(n) n 4 0 4  4 ×   ℇ (K × ) = 8n(n − 1) + 8n(n − 1) S(K1,n-1 )  n  n n 4 4 0  4 ℇ (K × ) = 16n(n − 1)         Denition 2.2:   Crown graph S for an integer n ≥ 2 is the graph with vertex n 4 4 4  0   set *u , u , … u , v , v , … v + and edge set *u v : 1 ≤ i, j ≤ n, i ≠ Characteristics polynomial is (−1) (λ + 4) [λ − 4(n − 2) − j+. n (n − 1)] Characteristics equation is (−1) (λ + 4) [λ − 4(n − 2) − Theorem 2.11 For n ≥ 2, the Schultz energy of the crown graph is n (n − 1) = 0 ] 8 if n = 2 { 0 (2n − 4) ± √n + 3n − 16n + 16 ( ) Spec(K , )=( ) 8n n − 1 if n > 2 n − 2 1

Proof The Schultz matrix of crown graph isS(S ) =

The schultz energy is,

ℇ (K , ) = |−4|(n − 2) + |(2n − 4) + √n + 3n − 16n + 16 | (1) + |(2n − 4) − √n + 3n − 16n + 16 | (1)

ℇ (K , ) = 4(n − 2) + (2n − 4) + √n + 3n − 16n + 16 + √n + 3n − 16n + 16 − (2n − 4) Characteristic polynomial is λ ,λ − 6n(n − 1)-,λ − 2(n − 1)(n − 4)-,λ + 8(n − 1)-. ℇ (K , ) = 4(n − 2) + √n + 3n − 16n + 16 Characteristic equation is λ ,λ − 6n(n − 1)-,λ − 2(n − 1)(n − Case 2: if n=2 Then Schultz matrix of is, 4)-,λ + 8(n − 1)- = 0. Spec(S ) = 2 0 6n(n 1) 0 2(n 1)(n  4) 8(n 1) S(K , ) = . /   0 2 ×   1 n 1 1 1 Characteristic polynomial is (λ − 4)   Characteristic equation is (λ − 4) = 0 The Schultz energy is, 2 ℇ (S ) = |6n(n − 1)|(1) + |0|(n − 1) + |2n(n − 1)(n − 4)|(1) Spec(K , ) = . / 2 + |−8(n − 1)|(1) The Schultz energy is ,ℇ (K , ) = |2|(2) = 4 ℇ (S ) = 6n(n − 1) + 2(n − 1)(n − 4) + 8(n − 1)

ℇ (S ) = 8n(n − 1). Defination 2.1.cocktail party graph is denoted by Knx2, is a Case 2: if n=2 graph having the vertex set Then Schultz matrix crown graph is, V = ⋃ *u , v + and the edge set E=*u u , v v : i ≠ j+ ∪ 0 0 0 2 *u u , v v : 1 ≤ i < j ≤ n+.   Theorem 2.10.The Schultz energy of cocktail party graph is 0 0 2 0 16n(n-1) 0 2 0 0   Proof Case 1: ifn > 2   2 0 0 0 Consider cocktail party graph K × with vertex set V = S(S ) = Characteristic polynomial is ( ) ( ) ⋃ *u , v +. λ − 2 λ + 2 Characteristic equation is ( ) ( ) Then the schultz matrix of cocktail party is S(K × ) = λ − 2 λ + 2 = 0 2 −2 Spec(S ) = . / 2 2

The Schultz energy is ,ℇ (S ) = |2|(2) + |−2|(2) = 8 Definition 2.3. Friendship graph is the graph obtained by taking n copies of the cycle graph C3with a vertex in common.

It is denoted by F . Friendship graphF contains2n + 1 vertices and 3n edges.

Theorem 2.12. The Schultz energy of friendship graph ℇ (F ) is equal to4[(2n − 4) + √n + 3n − 16n + 16 ]. Characteristic polynomial is λ ,λ − 8n(n − 1)-,λ + 8(n − 1)-.

, -, - Characteristic equation is λ λ − 8n(n − 1) λ + 8(n − 1) = Proof 0. 28 IJSTR©2019 www.ijstr.org INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 8, ISSUE 11, NOVEMBER 2019 ISSN 2277-8616

For a friendship graph F with vertex set V = *v , v , v , … v , v , v … v +.

Then the Schultz matrix of friendship graph is ℇ (F ) = The Schultz energy is,

ℇ (K , ) = |−4n|(m − 1) + |−4m|(n − 1) + |,(4m − 2)n − 2m- + √mn + (2m + 4)n + (m − 8)mn + 4m | (1) + |,(4m − 2)n − 2m- − √mn + (2m + 4)n + (m − 8)mn + 4m | (1) Characteristic polynomial is (−1)(λ + 4) (λ + 12) ,λ − 4(4n − 3)λ − 8n(n + 1) - ℇ (K , ) Characteristic equation is (−1)(λ + 4) (λ + 12) ,λ − = 4,2mn − m − n- 4(4n − 3)λ − 8n(n + 1) - = 0. + |,(4m − 2)n − 2m- Spec(F ) + √mn + (2m + 4)n + (m − 8)mn + 4m | (1) + |,(4m − 2)n − 2m- − √mn + (2m + 4)n + (m − 8)mn + 4m | (1)

The Schultz energy is, CONCLUSIONS: ℇ (F ) = |−4|(n) + |−12|(n − 1) In this article we defined Schultz energy of a graph. Upper and ( ) √ ( ) + | 8n − 6 + 2 2n + 20n − 22n + 9| 1 lower bounds for Schultz energy are established. A − |(8n − 6) generalized expression for Schultz energies for star graph, complete graph, crown graph, complete bipartite graph,

− 2√2n + 20n − 22n + 9| (1) cocktail party graph and friendship graphs are also computed.

ℇ (F ) = 4n + 12(n − 1) + (8n − 6) + 2√2n + 20n − 22n + 9 + 2√2n + 20n − 22n + 9 − (8n − 6) AUTHORS CONTRIBUTIONS: All the authors worked together for the preparation of the √ ℇ (F ) = 4(4n − 3) + 2n + 20n − 22n + 9 manuscript and all of us take the full responsibility for the content of this article. However second author typed the article Theorem 2.13 The Schultz energy of the complete bipartite and all of us read and approved the final manuscript. graph is4,2mn − m − n- + CONFLICT OF INTERESTS: |,(4m − 2)n − 2m- + The authors hereby declares that there are no issues √mn + (2m + 4)n + (m − 8)mn + 4m | + |,(4m − 2)n − regarding the publication of this article.

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