Randic type additive connectivity energy of a graph
Автор: Madhusudhan Krishnarajapete Venkatarama, Reddy Polaepalli Siva Kota, Rajanna Karpenahalli Ranganathappa
Журнал: Владикавказский математический журнал @vmj-ru
Статья в выпуске: 2 т.21, 2019 года.
Бесплатный доступ
The Randic type additive connectivity matrix of the graph G of order n and size m is defined as RA(G)=(Rij), where Rij=√di+√dj 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.
Randic type additive connectivity energy, randic type additive connectivity eigenvalues
Короткий адрес: https://sciup.org/143168795
IDR: 143168795 | DOI: 10.23671/VNC.2019.2.32113
Текст научной статьи 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,..., mr. 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)2a2 = ^2 ai aij = 52 aiiajj - ajiaij = 52 aiiajj - 52 'A'A = -K >
1gi
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 Ai, 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 Kn is RED (Kn) = 4(n - 1)2.
<1 Lel, Kn be the complete gr;rph with vertex set V = {v1, v2,..., vn} 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, ••• , un,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, • • • , un, v1, v2, • • • , vn} is
RAE(Km,n ) = 2( V mn)( Vm + Vn)-
<1 Let Km,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(Km , 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 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 Kn of the complete graph Kn is
RAE(Kn = 0.
-
<1 Lel, Kn Ire the complete gnrph with vertex set V = {v1, v2,..., vn}. 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 KnX2 of the cocktail party graph Knx2 of оrder 2n is
RAE(Kn X 2^ = 4n.
-
< Let Kn X 2 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
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.
Список литературы Randic type additive connectivity energy of a graph
- Gutman, I. The Energy of a Graph, Ber. Math. Stat. Sekt. Forschungsz. Graz, 1978, vol. 103, pp. 1-22.
- 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.
- Prakasha, K. N., Siva Kota Reddy, P. and Cangul, I. N. Minimum Covering Randic Energy of a Graph, Kyungpook Math. J., 2017, vol. 57, no. 4, pp. 701-709.
- Siva Kota Reddy, P., Prakasha, K. N. and Siddalingaswamy, 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.
- 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.