Randic type additive connectivity energy of a graph Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2019, Volume 21, Issue 2, P. 18 26

УДК 519.17

DOI 10.23671/VNC.2019.2.32113

RANDIC TYPE ADDITIVE CONNECTIVITY ENERGY OF A GRAPH

K. V. Madhusudhan1, P. Siva Kota Reddy2 and K. R. Rajanna3

1 ATME College of Engineering, Mysore 570 028, Karnataka, India; 2 Siddaganga Institute of Technology, B. H. Road, Tumkur 572 103, Karnataka, India;

3 Acharya Institute of Technology, Bangalore 560 107, Karnataka, India E-mail: E-mail: kvmadhul3@gmail.com; reddy_math@yahoo.com, pskreddy@sit. ac . in;

raj anna@acharya.ac.in

Abstract. The Randic type additive connectivity matrix of the graph G of order n and size rn is defined as RA(G) = (Rij), where Rij = \/di+ \fd~j if the vertices Vi and Vj are adjacent, and Rij = 0 if Vi and Vj are not adjacent, where di and dj be the degrees of vertices vi and vj respectively. The purpose of this paper is to introduce and investigate the Randic type additive connectivity energy of a graph. In this paper, we obtain new inequalities involving the Randic type additive connectivity energy and presented upper and lower bounds for the Randic type additive connectivity energy of a graph. We also report results on Randic type additive connectivity energy of generalized complements of a graph.

Key words: Randic type additive connectivity energy, Randic type additive connectivity eigenvalues. Mathematical Subject Classification (2010): 05C50.

For citation: Madhusudhan, K. V., Reddy, P. S. K. and Rajanna, K. R. Randic Type Additive Connectivity Energy of a Graph, Vladikavkaz Math. J., 2019, vol. 21, no. 2, pp. 18-26. DOI: 10.23671/VNC.2019.2.32113.

1. Introduction

Let G be a simple, finite, undirected graph. The energy E(G) is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. Basically energy of graph is originated from chemistry. In For more details on energy of graphs (see fl, 2]).

In chemistry we can represent the conjugated hvdrocarbos by a molecular graph. Each edge between the carbon-carbon atoms can be represented by an edge. Here we will neglect the hydrogen atoms. Now a days energy of graph attracting more and more researchers due its significant applications. The Randic type additive connectivity matrix RA(G) = (Rij)nxn is given by

RAi- = +

ij 1 0, otherwise.

The characteristic polynomial of RA(G) is denoted by <fiRA(G,\) = det(A/ — RA(G)). Since the Randic type additive connectivity matrix is real and symmetric, its eigenvalues are

© 2019 Madhusudhan, К. V., Reddy, P. S. K. and Rajanna, K. R.

real numbers and we label them in non-increasing order Ai > A2 > ■ ■ ■ > An. The minimum dominating Randic energy is given by

n

RAE(G) = J] |Ai(1)

i=1

Definition 1.1. The spectrum of a graph G is the list of distinct eigenvalues A1 > A2 > ■ ■ ■ > Ar, with their multiplicities m1, m2,..., mr, and we write it as

Spec(G) = (Ai A2 ■■■ M .

ym1 m2 ■ ■ ■ mrJ

In [3, 4], the authors defined the minimum covering Randic energy of a graph and minimum dominating Randic energy of a graph and presented the upper and lower bounds on these new energies.

This paper is organized as follows. In the Section 3, we get some basic properties of Randic type additive connectivity energy of a graph. In the Section 4, Randic type additive connectivity energy of some standard graphs are obtained.

2. Some basic properties of Randic type additive connectivity energy of a graph

Let us define the number K as

i<j

Then we have

Proposition 2.1. The first three coefficients of the polynomial 0ra(G, A) are as follows:

(i) ao = 1,

(ii) a1 = 0,

(iii) a2 = -K.

< (i) By the definition of $ra(G, A) = det[A1 — RA(G)], we get a0 = 1.

(ii) The sum of determinants of all 1 x 1 principal submatrices of RA(G) is equal to the trace of RA(G) implying that

a1 = (—1)1 x the trace of RA(G) = 0.

(iii) By the definition, we have

(- 1)2a2 = aai. = aiiajj— ajiaij = aiiajj— ajiaij = —K.>

Proposition 2.2. If Ai, A2,..., An are the Randic type additive connectivity eigenvalues of RA(G), then

X>2 = 2K.

< It follows as

n n n n

E A2 = E E aijaji = ^(aij)2 + 53(a*)2 = 2 53(ay-)2 = 2P. >

i=1 i=1 j=1 i<j i=1 i<j

Using this result, we now obtain lower and upper bounds for the Randic type additive connectivity energy of a graph:

Theorem 2.1. Let G be a graph with n vertices. Then

RA(G) < V2nK.

< A1, A2, . . . , An be the eigenvalues of RA(G). By the Cauchy-Schwartz inequality we have

/n \ 2 / n \ / n \

22

53*6* ^ 53

ai

i=1 i=1 i=1

Let ai = 1 bi =| Ai |. Then

(n \ 2 / n \ / n \

< (§V fe^2)

implying that

[RAE]2 ^ n ■ 2K

and hence we get _

[RAE] ^ V2nK

as an upper bound. >

Theorem 2.2. Let G be a graph with n vertices. If R= det RA(G), then

RAE(G) > \JlK + n{n- l)Rn.

< By definition, we have

(n \ 2 n n / n \

£| Ai | = E | Ai | 53 | Aj |= | Ai |2 + E | Ai || Aj | .

i=1 i=1 j=1 i=1 i=j

Using arithmetic-geometric mean inequality, we have

t ^ n(n —1)

E| Ai || Aj | > m | Ai || Aj |

n(n - 1) .= . = .

i=j i=j

Therefore,

n / \ «(«-!) 2 ^ V^ | A 12 ...................^

[RA(G)]2 | Ai |2 +n(n - 1) [ft | Ai || Aj |

i=1 i=j

i

n(n — 1)

| Ai |2 +n(n - 1)(n | Ai |2(n-1)'

i=1 i=1 n

= E I Ai 12+n(n-l)R« =2K+ n(n-l)R

i

i

Thus,

RAE(G) ^ \jlK + n{n- l)Rn. o

Let An and Ai are the minimum and maximum values of all Ais. Then the following results can easily be proven by means of the above results:

Theorem 2.3. For a graph G of order n,

n2

RAE(G) > \\2Kn - — (Ai - Ara)2. Gn

2v/A^V2 Kn

RAE(G) ^

(Ai + A„)2

Theorem 2.5. Let G be a graph of order n. Let Ai ^ A2 ^ A3 ^ ... ^ An be the eigenvalues in increasing order. Then

|Ai| + |An|

3. Randic type additive connectivity energy of Some Standard Graphs

Theorem 3.1. The Randic type additive connectivity energy of a complete graph Kn is RED{Kn) =4(n-l)i.

< Let Kn be the complete graph with vertex set V = {vi,v2,... ,vn}. The Randic type additive connectivity matrix is

RA(Kn) =

1_ 2y/nTrT 2y/n^T

2y/n-l 2Vn - 1 0

2^n - 1 2^n - 1 2^/n^l 2^/n^l 2 Vn - 1 2 y/n - 1

2y/n-l 2Vn - 1

2y/n-l 0 2^/n - 1

Hence, the characteristic equation is

and the spectrum is

(A + 2V^T)ra"i(A-2(n-l)i) =0

Spec DR(Kn) = (2in~1)l

V 1 n — 1 J

Therefore, we get RAE(Kn) = 4(n — 1)2. >

Theorem 3.2. The Randic type additive connectivity energy of star graph K\n-1 is

RAE(Klin-i) = 2 [Vn-1 + {n- 1)].

< Let K1jn-1 be the star graph with vertex set V = {v0, v1,..., vn-1}. Here v0 be the center. Randic type additive connectivity matrix is

RA(K1,n-1) =

1

y/n- 1 + 1 y/n- 1 + 1

Vn^T+1 V^T + l

Vn^1 + 1

0 0

0 0

y/n - 1 + 1 y/n-1 + 1'

0 0

0 0

The characteristic equation is

\n~2(A + y/n^l + (n — 1)) (A — (v^T+(n-l))) =0 and the spectrum would be

y/n-1+ (n-l) 0 -y/n - 1 + (n - 1)

Specg (K1,n-1) =

n2

Therefore, RAE(Kitn-i) = 2[y/n^l + (n - 1)]. >

Theorem 3.3. The Randic type additive connectivity energy of Crown graph JS

RAE(S°n) = 8(n- 1)1

< Let Sn be a crown graph of order 2n with vertex set {u1,u2, ■■■ ,un,v1,v2, ■■■ ,vn}. The Randic type additive connectivity matrix is

RAE(Sn) =

0 _

0 2y/n^l

2y/n^l 0

2y/n^l 2yfn^l

0 0_ 2y/n^T

0 2yfn^l 0

0 2yfn^l

0 2y/n^T

2V/n^T 1

2yfn^l 0

0

2y/n^l

Hence, the characteristic equation is

(A + 2y/^l)n~1 (A - 2y/^l)n~1 (A - 2(n - 1)1) (A + 2(n - 1)1) = 0 and spectrum is

SpecflA(5S)=f 2(n"1)§ -2(n"1)f "2^TV

1 1 n- 1 n- 1

'n- 1

<n- 1

n- 1

0

0

0

0

0

Therefore, RAE(S°n) = 8(n - 1)3. >

1

Theorem 3.4. The Randic type additive connectivity energy of complete bipartite graph Kn,nof ord er 2n with vertex set {ui,u2, • • • , un, vl,v2, ••• , vn} is

RAE(Km,n) = 2{y/rrm){y/m + y/n).

< Let Km,n be the complete bipartite graph of order 2n with vertex set {ul,u2, • • • , un, vl, v2, • • • , vn}. The Randic type additive connectivity matrix is

R (Km,n) —

0 0 0 0

m + y/n

rn + yß m + yß

0 0 0 0

m + y/n

m + y/n m + y/n

0 0 0 0

m + y/n m + y/n m + y/n

m + y/n m + m + m +

m + m + y/n m + m + y/n

0 0 0

0 0 0

m + V n m + m + m +

0 0 0

Hence, the characteristic equation is

Ara_2[A — {y/rrm){y/m + \/n)][A + {y/rrm){y/m + y^-)] =0. Hence, spectrum is

(y/mn)(y/m + y/n) 0 ~(y/mn)(y/m + y7^)]

SPecRA (Km,n) —

1

1

m + n — 2

Therefore, RAE{Km>n) = 2{y/nm)(y/m + y/n). >

Theorem 3.5. The Randic type additive connectivity energy of Cocktail party graph Knx2 is

RAE(K,nx2

An- 6

n — 1

< Let Knx2 be a Cocktail party graph of order 2n with vertex set

{Ui,U2, . . . ) un ,Vi,V2, Vn }• The Randic type additive connectivity matrix is

0 2 A/2«. - 2 2 A/2« - 2 . 0 2 A/2 « - 2 2 A/2« - 2 2 A/2« - 2

2 a/2n - 2 0 2y/2n - 2 . . 2 A/2« - 2 0 2y/2n - 2 2y/2n - 2

2y/2n - 2 2 A/2« - 2 0 . 2y/2n - 2 2 A/2« - 2 0 2y/2n - 2

2y/2n - 2 2y/2n - 2 2 A/2« - 2 . . 2 A/2 « - 2 2 A/2« - 2 2 A/2« - 2 0

RA(Knx 2) —

0 2 a/2 « - 2 2 a/2« - 2 . 0 2 a/2« - 2 2 a/2« - 2 2 a/2« - 2

2 a/2«- - 2 0 2y/2n - 2 . . 2 a/2« - 2 0 2y/2n - 2 2 a/2 « - 2

2y/2n - 2 2 a/2« - 2 0 . 2y/2n - 2 2 a/2 « - 2 0 2a/2 « - 2

2y/2n - 2 2y/2n - 2 2 a/2« - 2 . . 2y/2n - 2 2y/2n - 2 2 a/2« - 2 0

Hence, the characteristic equation is

Xn(X + 4V2n-2)n (A — 4(n — l)y/2n — 2) =0

and the spectrum is

SpecRA (Knx2) —

4(n — l)y/2n — 2 0 —4y/2n - 2 1 n n — 1

Therefore, RAE(Knx2) = S(n - l)y/2n-2. >

4. Randic type additive connectivity energy of complements

Definition 4.1 [5]. Let G be a graph and Pk = {Vi, V2,..., Vk) be a partition of its vertex set V. Then the fc-complement of G is denoted by (G)fc and obtained as follows: For all V^ and Vj in i = j remove the edges between and Vj and add the edges between the vertices of Vi and Vj which are not in G.

Definition 4.2 [5]. Let G be a graph and Pk = {Vi, V2, • • •, Vk} be a partition of its vertex set V. Then the fc(i)-complement of G is denoted by (G)k^ and obtained as follows: For each set V^ in P^, remove the edges of G joining the vertices within Vr and add the edges of G (complement of G) joining the vertices of Vr.

Here we investigate the relation between some special graph classes and their complements in terms of the Randic type additive connectivity energy.

Theorem 4.1. The Randic type additive connectivity energy of the complement of the complete graph is

RAE{Kn) = 0.

< Let be the complete graph with vertex set V = (vi, ... ,vn}. The Randic type additive connectivity matrix of the complement of the complete graph is

RA(Kn) =

Characteristic polynomial is

RA(Kn) =

0 0 0 . .0 0

0 0 0 . .0 0

0 0 0 . .0 0

0 0 0 . .0 0

0 0 0 . .0 0

A 0 0. . . 0 0

0 A 0. . . 0 0

0 0 A. . . 0 0

0 0 0. . . A 0

0 0 0. . . 0 A

Clearly, the characteristic equation is An = 0 implying

RAE(K= 0. >

Theorem 4.2. The Randic type additive connectivity energy of the complement Knx2 of the cocktail party graph Knx2 of order 2n is

RAE(K„x2) = 4n.

< Let Knx2 be the cocktail party graph of order 2n having the vertex set (u1,u2, ■ ■ ■ , un, v1,v2, ■ ■ ■ , vn}. The corresponding Randic type additive connectivity matrix

is

RA(Knx 2)

Characteristic polynomial is

"0 0 0 0 . .2 0 0 0

0 0 0 0 . .0 2 0 0

0 0 0 0 . .0 0 2 0

0 0 0 0 . .0 0 0 2

2 0 0 0 . .0 0 0 0

0 2 0 0 . .0 0 0 0

0 0 2 0 . .0 0 0 0

0 0 0 2 . .0 0 0 0

RA(K,

nx 2) —

A 0 0 0 .. . -2 0 0 0

0 A 0 0 .. .0 -2 0 0

0 0 A 0 .. .0 0 -2 0

0 0 0 A .. .0 0 0 -2

-2 0 0 0 .. .A 0 0 0

0 -2 0 0 .. .0 A 0 0

0 0 -2 0 .. .0 0 A 0

0 0 0 -2 . . .0 0 0 A

and the characteristic equation becomes

(A + 2)n(A - 2)n = 0 implying that the spectrum would be

SpecRA (Knx2) = ^ 2

Therefore,

RAE(Knx2) = 4n. O

Acknowledgement. The authors are thankful to the anonymous referee for valuable suggestions and comments for the improvement of the paper.

References

1. Gutman, I. The Energy of a Graph, Ber. Math. St,at. Sekt. Forschungsz. Graz, 1978, vol. 103, pp. 1-22.

2. Gutman, I. The Energy of a Graph: Old and New Results, Algebraic Combinatorics and its Applications / eds. Betten, A., et al., Berlin, Springer-Verlag, 2001, pp. 196-211.

3. Prakasha, K. N., Siva Kot,a Reddy, P. and Cangül, I. N. Minimum Covering Randic Energy of a Graph, Kyungpook Math. J., 2017, vol. 57, no. 4, pp. 701-709.

4. Siva Kota Reddy, P., Prakasha, K. N. and Siddaiingaswamy, V. M. Minimum Dominating Randic Energy of a Graph, Vladikavkaz. Math,. J., vol. 19, no. 1, pp. 28-35. DOI 10.23671/VNC.2017.2.6506.

5. Sampathkumar, E., Pushpalatha, L., Venkatachalam, C. V. and Pradeep Bhat, Generalized Complements of a Graph, Indian J. Pure Appl. Math,.., 1998, vol. 29, no. 6, pp. 625-639.

Received September 7, 2018

Krishnarajapete Venkatarama Madhusudhan

ATME College of Engineering,

Mysore 570 028, Karnataka, India,

Assistant Professor

E-mail: kvmadhul3@gmail.com;

Polaepalli Siva Кота Reddy Siddaganga Institute of Technology, В. H. Road, Tumkur 572 103, Karnataka, India, Associate Professor

E-mail: reddy_math@yahoo.com, pskreddy@sit. ac . in;

Karpenahalli Ranganathappa Rajanna Acharya Institute of Technology, Bangalore 560 107, Karnataka, India, Professor and Head of Mathematics E-mail: rajanna@acharya. ac. in

Владикавказский математический журнал 2019, Том 21, Выпуск 2, С. 18^26

ЭНЕРГИЯ АДДИТИВНОЙ СВЯЗНОСТИ ТИПА I'Л1 ЦИКЛ ГРАФА

К. В. Мадхусудхан1, П. Сива Кота Редди2, К. Р. Раджанна3

1 Инженерный колледж, Майсур 570 028, Картанака, Индия;

Сиддаганга технологический институт, Тумкур 572 103, Картанака, Индия;

3 Технологический институт Ачарьи, Бангалор 560 107, Карнатака, Индия E-mail: kvmadhul3@gmail.com; reddy_math@yahoo.com, pskreddy@sit. ас . in; га j anna@achary а. ас . in

Аннотация. Матрица аддитивной связности типа Рандика RA(G) = (Rij)nxm задается равенствами Rij = \fdi + y/dj, если вершины Vi и Vj смежны, Rij = 0, в противном случае, где di и dj — степени вершин «¿и Vj соответственно. Целью данной статьи является исследование энергии аддитивной связности типа Рандика. В данной статье мы получили новые неравенства, включающие энергию аддитивной связности типа Рандика, и представили ее верхнюю и нижнюю границы. Мы также получили результаты по энергии аддитивной связности типа Рандика обобщенных дополнений графа.

Ключевые слова: энергия аддитивной связности типа Рандика, собственные значения аддитивной связности типа Рандика.

Mathematical Subject Classification (2010): 05С50.

Образец цитирования: Madhusudhan К. V., Reddy P. S. К. and Rajanna К. R. Randic Type Additive Connectivity Energy of a Graph // Владикавк. мат. журн.—2019.—Т. 21, № 2.—С. 18-26 (in English). DOI: 10.23671/VNC.2019.2.32113.