Randic type additive connectivity energy of a graph

Lot 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 [1, 2]).

In chemistry, we can represent the conjugated hydrocarbos 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

RAij = Г^+Vd’ ’

v i ∼ v j , otherwise.

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

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

n

RAE(G) = £|Ai |.                               (1)

i=1

Definition 1.1. The spectrum of a graph G is the list of distinct eigenvalues Ai > A2 > • • • > Ar. with their 1nultiplicitics m1,m₂,..., m_r. and we write it as

Spee(G) = fA1   A2 ••• Ar ) .

mi m2 • • • mr

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

K = E(Vdi + Vdj Л i

Then we have

Proposition 2.1. The first three coefficients of the polynomial фRA(G, A) are as follows: (i) ao = 1.

• (ii)    a1 = 0.

• (iii)    a2 = —K.

• <1    (i) By the clefmitioii of ФRA(G, A) = det[AI — RA(G)]. we get ao = 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 tr;ree of RA(G) = 0.

• (iii)    By the definition, we have

^{( —1)2}a2 = ^2    ai  aij = 52 aiiajj - ajiaij = 52 aiiajj - 52 'A'A = -K ^>

1gij   jj   1^i

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

£ Ai2 = 2K.

i=1

<1 II, follows as

n    nn              n

E A2 = E E a ij a ji = ² E^' )2 + ТЫ = 2 E^')2 = 2P ▻ i=i i=i j=i            i

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) a V2nK.

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

( n \ ²    / n A / n A

E«л) a (E^(Eb i ²)-

Lol, ai = 1. bi =| Ai |. Thon

(n \ 2    / n \ / n \ ew)a e1 И

implying that

[RAE]² a n • 2K

and hence we get

[RAE] a V2nK

as an upper bound. >

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

RAE(G) >   2K + n(n — 1)R 2 .

• <    By definition, we have

( n      \ ² n        n          / n       \

Ei A i l =ElA i lElA j l = ElA i l²  +ElA i l|A j |.

i =i              i =i        j =i            i =i              i = j

Using arithmetic-geometric mean inequality, we have

—-—- V| A i | | A j | >  n(n — 1) ²—-''         ³

(        ) i=j

|λi ||λj | i=j

Therefore,

[RA(G)]2

>

>

n

E l A i l² +n(n — i =i

n

E l A i l² +n(n — i =i

1 n ( n- 1)

1 n ( n- 1)

1) I П l A i ll A j l i = j

n

| λi i=i

| 2(п - 1)^

1 n ( n- 1)

= ^E l A i l² +n(n — 1)R² =2K + n(n — 1)R 2 . i =i

Thus,                                   _________________

RAE(G) >  2K + n(n - 1)R2. ▻

Let An and A1 are the minimum and maximum values of all A’s. Then the following results can easily be proven by means of the above results:

Theorem 2.3. For a graph G of order n,

RAE(G) >

n2

у 2Kn--4(A1 - An)2.

Theorem 2.4. For a graph G of order n with non-zero eigenvalues, we have

RAE(G) >

2 хЛА V2Kn

^(A1 ⁺ Xn)²

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

RAE(G) >

|Ai||An |n + 2K |A1| + | An |

• 3.    Handle type additive connectivity energy of Some Standard Graphs

Theorem 3.1. The Handle type additive connectivity energy of a complete graph K_n is RED (K_n) = 4(n - 1)2.

<1 Lel, Kn be the complete gr;rph with vertex set V = {v₁, v₂,..., v_n} The Randic type additive connectivity matrix is

	' 1	2 Vn - 1	2 Vn - 1	...      2 Vn - 1			2 Vn - 1
	2V n - 1	0	2Vn - 1	...	2 Vn -	1	2 Vn -	1
RA(K n ) =	2Vn - 1 ■ ■ ■	2Vn - 1 * * *	0 * * *	... * * *	2Vn - * * *	1	2Vn - * * *	1
	2Vn - 1	2Vn - 1	...	2Vn - 1	0		2Vn -	1
	_2V n - 1	2Vn - 1	...	2Vn - 1	2Vn -	1	0	-
Hence, the characteristic equation is

(A + 2V n - 1 )^{n 1}(A - 2(n - 1) 2) =0

and the spectrum is

SpecD (Kn) = f 2(n - 12 -2Vn-T A . 1 n-1

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

Theorem 3.2. The Randic type additive connectivity energy of star graph K₁,_n-1 is

RAE(K1n—1) = 2 [Vn-T + (n - 1)].

<1 Let K1,n-1 be the star graph with vertex set V = {vo, v1,..., vn-1}. Here vo be the center. Randic type additive connectivity matrix is

RA(K1 ,n -1) =	1 Vn — 1 + 1		V n — 1 + 1 0 0 * * *	V n — 1 + 1 0 0 * * *	. . . Vn — 1 + 1 ...         0 ...         0 .                                           ■ •                                      .	V n — 1 + 1 0 0 * * *
	n	—1 + 1 * * *
	n	—1 + 1	0	0	...         0	0
	n	—1 + 1	0	0	...         0	0

The characteristic equation is

A n ^-2 (A + Vn — 1 + (n — 1)) (A — (Vn — 1 + (n — 1))) = 0

and the spectrum would be

Spec R (Ki ,n -1) = ( ^n —¹ +⁽n - ^l) n ⁰

-Vn — 1 + (n — 1)

.

Therefore. RAE(Ki,_{n -} i) = 2[V n — 1 + (n — 1)]. ▻

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

RAE(Sn ) = 8(n — 1) 2.

< Lel, Sn Ire a crown graph of order 2n with vertex set {u1,u₂, ••• , u_n,v1,v₂, ••• , vn}. The Randic type additive connectivity matrix is

0         0      ...      0         0      2 Vn — 1     ...     2 Vn — 1" 0         0     ...     0      2Vn — 1     0        ...      2Vn — 1 0        0     ...     0     2Vn — 1    ...     2Vn — 1 2Vn — 1 •                                        •                         .                                •                                        •                                        •                                    .                                           •

RAE(Sno ) =

•                                        •                            •                            •                                        •                                        •                                        •                                        • .                                        .                                •                         .                                        .                                        .                                           •                                    . 0         0     ...     0     2Vn — 1     ...      2Vn — 1     0 0     2Vn — 1 ...   2Vn — 1     1         0        ...        0 2Vn — 1     0     ...   2Vn — 1     0         0        ...        0 •                                        •                         .                                •                                        •                                        •                                    .                                           • •                                        •                            •                            •                                        •                                        •                                        •                                        • .                                        .                                •                         .                                        .                                        .                                           •                                    . 2Vn — 1  2Vn — 1  ...  2Vn — 1     0        0        ...        0 _ 2Vn — 1  2Vn — 1  ...      0         0         0         ...         0   _

Hence, the characteristic equation is

(A + 2Vn—T) n 1 (A — 2Vn—T) n 1 (A — 2(n — 1)2) ^A + 2(n — 1)2) = 0

and spectrum is

2(n — 1) 2

—2(n — 1) 2 2Vn—T

1 n—1

—2 Vn—T

n-

.

Therefore. RAE(S n ) = 8(n — 1) 2. ▻

Theorem 3.4. The Handle type additive connectivity energy of complete bipartite graph

Kn,nof order 2n with vertex set {u1, u₂, • • • , u_n, v1, v₂, • • • , vn} is

RAE(Km,n ) = 2( V mn)( Vm + Vn)-

<1 Let K_m,_n be the complete bipartite graph of order 2n with vertex set

{u1, u2, • • • , un, v1, v2, • • • , vn}. The Randic type additive connectivity matrix is

0       0       0    --- mm + Vn mm + nm mm + nm 0       0       0    --- mm + nn mm + nn mm + Vn 0       0       0    --- mm + vn Vm + Vn Vm + Vn

R D (K m,n ) =

0       0       0    --- vm + Vn Vm + Vn Vm + Vn •                                                          •                                                          •                                   .                                            •                                                          •                                                          • •                                              •                                              •                                •                                •                                              •                                              • •                                              •                                              ■                                                                 ■                                              •                                              • Vm + Vn Vm + Vn Vm + Vn ---    0       0       0 Vm + Vn Vm + Vn Vm + Vn ---    0       0       0 Vm + Vn Vm + Vn Vm + Vn ---    0       0       0

Hence, the characteristic equation is

A n ²[A - (Vmn)(Vm + Vn)][A + (Vmn)(Vm + Vn)] = 0.

Hence, spectrum is

0 m+n-

—(Vmn)(Vm + Vn)]

(Vmn)(Vm + Vn)

SpeC RA ⁽K m,n ) =           1

Therefore, RAE(K_m , _n) = 2(Vmn)(Vm + Vn)- ^

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

K n x2 R

.

4n

RAE(K n _X2) = — n

-

-

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

{ui, U2, - - -, un, v1, v2, - - -, vn}• The Randic type additive connectivity matrix is

		0     2V2n — 2	2V2n	2 - - -	0		2V2n	— 2	2V2n —	2	2V2n —	2
		2V2n — 2     0	2V2n	2 - - -	2V2n —	2	0		2V2n —	2	2V2n —	2
		2V2n — 2 2V2n — 2	0	...	2V2n —	2	2V2n	— 2	0		2V2n —	2
		2V2n — 2 2V2n — 2	2X 2n	—2 ---	2V2n —	2	2V2n	— 2	2V2n —	2	0
RA(KnX2)	=	•                                         ■ •                                         ■ •                                         ■	* ♦ ♦		♦ ♦ ♦		♦ ♦ ♦		♦ ♦ ♦		♦ ♦ ♦
		0     2V2n — 2	2V2n	—2 ---	0		2V2n	— 2	2V2n —	2	2V2n —	2
		2V2n — 2     0	2V2n	—2 ---	2V2n —	2	0		2V2n —	2	2V2n —	2
		2V2n — 2 2V2n — 2	0	...	2V2n —	2	2V2n	— 2	0		2V2n —	2
		2V2n — 2 2V2n — 2	2 V 2 n	—2 ---	2V2n —	2	2V2n	— 2	2V2n —	2	0
Hence,	the characteristic equation is			1(A — 4(
		A n (A + 4V2n	— 2) n"		n — 1)V	2n	—2) =	0
and the spectrum is			^ 4(n						)-
		SPec RA (K n x2) =		— 1)V2n 1	— 2 0 n	-	4V2n -n—1	- 2

Therefore. RAE(KnX2) = 8(n — 1)V2n — 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 k-complement of G is denoted by (G)k and obtained as follows: For all Vi and Vj in Pk, i = j, remove the edges between Vi and Vj and add the edges between the vertices of Vi arid Vj which ai-e not in G.

Definition 4.2 [5]. Let G Ire a, graph and Pk = {V₁, V₂,..., Vk } be a partition of its vertex set V. Then the k(i)-complement of G is denoted by (G)k(i) and obtained as follows: For each set Vr i 11 Pk. remove tlie edges of G joining the vertices within V_r 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 Kn of the complete graph Kn is

RAE(Kn = 0.

• <1 Lel, Kn Ire the complete gnrph with vertex set V = {v₁, v₂,..., v_n}. The Randic type additive connectivity matrix of the complement of the complete graph Kn is

0

0

0 .

.0

0

0

0

0 .

.0

0

RA(Kn) =

0

•

•

•

0

*

*

*

0   .

•             .

•

*

.0

*

■

•             .

0 ■

0

0

0 .

.0

0

0

0

0 .

.0

0

Characteristic polynomial is

λ

0

0.

.. 0

0

0

λ

0.

.. 0

0

RA(Kn) =

0

* *

*

0

* *

*

λ.

•

*

.. 0

*

■

• .

0

■ ■

•

0

0

0.

.. λ

0

0

0

0.

.. 0

A

Clearly, the characteristic equation is An = 0 implying

RAE(Kn) = 0. ▻

Theorem 4.2. The Randic type additive connectivity energy of the complement K_nX2 of the cocktail party graph K_nx2 of оrder 2n is

RAE(Kn _X 2^ = 4n.

• < Let K_{n X} 2 be the cocktail party graph of order 2n having the vertex set {u₁,u₂, ••• ,u_n,v₁,v₂, ••• , vn}. The corresponding Randic type additive connectivity matrix

  IS
  			Г0	0	0	0 ...	2	0	0	01
  			0	0	0	0 ...	0	2	0	0
  			0	0	0	0 ...	0	0	2	0
  			0	0	0	0 ...	0	0	0	2
  RA(K n X2) =			• • •	* * *	* * *	•             . •                  • ■	* * *	* * *	* * *	■ ■ ■
  			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
  Characteristic polynomial is
  	A	0		0		0   ...	-2		0	0		0
  	0	λ		0		0   ...	0		-2	0		0
  	0	0		λ		0   ...	0		0	-	2	0
  	0	0		0		λ ...	0		0	0		-2
  RA(K n x2) =	■ ■ ■			* * *		• .                      ■	* * *		* * *			■
  	-2	0		0		0   ...	λ		0	0		0
  	0	-	2	0		0   ...	0		λ	0		0
  	0	0		-2		0   ...	0		0	λ		0
  	0	0		0		-2 . . .	0		0	0		A

and the characteristic equation becomes

(λ+2)n(λ-2)n=0

implying that the spectrum would be

Spec RA (K n x2

)=

2 n

-2 n

.

Therefore,

RAE(K n _X2) = 4n. ▻

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