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Владикавказский математический журнал 2017, Том 19, Выпуск 2, С. 28-35
MINIMUM DOMINATING RANDIC ENERGY OF A GRAPH
P. S. K. Reddy, K. N. Prakasha, V. M. Siddalingaswamy
In this paper, we introduce the minimum dominating Randic energy of a graph and computed the minimum dominating Randic energy of graph. Also, obtained upper and lower bounds for the minimum dominating Randic energy of a graph.
Mathematics Subject Classification (2010): 05C50.
Key words: minimum dominating set, minimum dominating Randic eigenvalues, minimum dominating Randic energy.
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. For more details on energy of graphs (see [5, 6]).
The Randic matrix R(G) = (Rj)nxn is given by Bozkurt et al. [1-3].
Ri
_ ) y/didi 10,
if vi ~ v
3 J
otherwise.
We can see lower and upper bounds on Randic energy in [1, 2, 4]. Some sharp upper bounds for Randic energy of graphs were obtain in [3].
2. The Minimum Dominating Randic Energy of Graph
Let G be a simple graph of order n with vertex set V = [v\, v2, v3,..., vn} and edge set E. A subset D of V = V(G) is called a dominating set if every vertex in V — D adjacent to a vertex of D. Minimum dominating set is called a dominating set of minimum power.
DG Rd (G) = (RD)nxn is given by
rD =
Md7
vi
3
if i = j and vi G D, otherwise.
The characteristic polynomial of RD(G) is denoted by (G, A) = det(A/ — RD(G)). Since the minimum dominating Randic Matrix is real and symmetric, its eigenvalues are
© 2017 Reddy P. S. K., Prakasha K. N., Siddalingaswamy V. M.
1
1
rsj
real numbers and we label them in non-increasing order Ai > A2 > ... An. The minimum dominating Randic Energy is given by
n
RED(G) = £ |Ai|. (1)
i=1
Definition 2.1. 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 .
\m1 m2 ... mrJ
This paper is organized as follows. In the Section 3, we get some basic properties of minimum dominating Randic energy of a graph. In the Section 4, minimum dominating Randic energy of some standard graphs are obtained.
3. Some Basic Properties of Minimum Dominating Randic Energy of a Graph
Let us consider
1
p = T ■
^ didj i'<3 J
Where didj is the product of degrees of two vertices which are adjacent.
Proposition 3.1. The first three coefficients of ^D (G, A) are given as follows:
(i) ao = 1,
(ii) ai = —|D| ,
(iii) a2 = |D|C2 - P.
< (i) Prom the définition $D (G, A) = det[A1 — RD (G)], we get a0 = 1.
(ii) The sum of determinants of all 1 x 1 principal submatrices of RD (G) is equal to the trace of Rd (G).
^ ai = (—1)1 trace of [RD(G)] = —|D|.
(iii)
/ i\2 aii aij
(—1) a2 = > , = > , auajj — ajiaj
' aji ajj '
1<i<Kn
= aiiajj — aji aij = |D|C2 — P. >
Proposition 3.2. If A1, A2,..., An are the minimum dominating Randic eigenvalues of Rd (G), then
n
2
J2Xi 2 = |D| + 2P.
< We know that
n n n n
E A2 = E E j = 2 E(aj )2 + E^ = 2^2(aij )2 + |D| = |D| + 2P. >
i=1 i=1 j=1 i<j i=1 i<j
Theorem 3.1. Let G be a graph with n vertices and then
RED(G) < yJn{\D\+2[P}),
where
1
i<j j
didj'
for which didj is the product of degrees of two vertices which are adjacent.
< A1, A2, . . . , An be the eigenvalues of RD (G). Now by Cauchy-Schwartz inequality we have
(E«^J ^E^J (Ebi
Let ai = 1, bi =| Ai Then
/ n \ 2 / n \ / n
ElAiM < E0 EM
i2
1 / \i=w \i=1 /
D2
[RED]2 ^ n(|D| +2P), [REd] < y/n(\D\+2P),
which is upper bound. >
Gn If R = det Rd (G), then
REd{G) ^ ^/(|£>|+2P) + n(n-l)i?r
< By definition,
(n \ 2 n n / n \
El Ai l = E l Ai l £ l Aj |= £ | Ai l2 + £ | Ai || Aj |
i=1 / i=1 j=1 Vi=1 / i=j
Using arithmetic mean and geometric mean inequality, we have
t n(n-l)
E iA* ii ^ II iA* H i
n(n - 1)^'"*
i=j i=j
Therefore,
i
n
n(n— 1)
[RED (G)]2 > £ l Ai l2 +n(n - 1) If] l Ai ll Aj l i=1 \i=j
1
n(n— 1)
> Y, l Ai l2 +n(n - 1)(II l Ai l2(n-1)-i=1 \i=1 /
n
= E iA* i2 +n(-n -R" = (i^i +2p)+n{n -R
i=1
i
Thus,
REd{G) ^ \J(\D\ + 2P) + n(n — 1)RÎ. >
4. Minimum Dominating Randic Energy of Some Standard Graphs
Theorem 4.1. The minimum dominating Randic energy of a complete graph Kn is RED{Kn) = ^Et.
< Let Kn be the complete graph with vertex set V = {v1; v2,..., vn}. The minimum dominating set = D = {v1}. The minimum dominating Randic matrix is
RD(Kn) =
1
l l
n—1 n—1
0
1
n— 1 1 1 ri— 1 ri— 1
1
n— 1
1 1
ri— 1 ri— 1
1 1
n— 1 ri— 1
1 1
ri— 1 n— 1
1 1
ri— 1 ri— 1
1 1
■ n— 1 ri— 1
0
1
ri— 1 1 1 ri— 1 n— 1
1
ri— 1
Characteristic equation is
A +
n — 1
n-2
2 2n — ^ n — 3
A2
1A+ n 1 n 1
and the spectrum is
Specg (Kn) =
(2ra-3)+V4ra-3 (2ra-3)~V4ra-3
2(n-l) 1
2(n-l)
1
-1 ri—1
n2
Therefore, RED(Kn) = >
Theorem 4.2. The minimum dominating Randic energy of star graph Ki>n-i is
^(Ki.n.i) = V5.
< Let K1>n-1 ^e the star graph with vertex set V = (v1; v2,..., vn}^ere v0 be the center. The minimum dominating set = D = |vo}. The minimum dominating Randic matrix is
RD (Ki,n-i ) =
l
fn^-
/n—l l
n- i 0
0 0
1
/n— 0
/n—l 0
0 0
_J_
/n—l 0
0
0 0
1
0
i
i
Characteristic equation is
(A)n-2[A2 - A - 1] =0
( 1+V5 Q 1—л/5 \
spectrum is Spec™ (Ki„_i) = 2 2 Therefore, RED{Kin_i) = y/b. >
\ 1 n — 2 1/
Теорема 4.3. The minimum dominating Randic energy of Crown graph SП is
red(s0) = (4n - 7) + V4n2 - 8n + 5
< Let Sn be a crown graph of order 2n with vertex set {u1, u2,..., un, v1, v2,..., vn} and minimum dominating set = D = {u1 ,v1}. The minimum dominating Randic matrix is
(Sii) =
1 0 0
0
0 l
n— 1
0 0 0
0 1
n— 1 0
1 1
га—1 ra—1 1 1
■ n— 1 ra—1
0 0 0
0 1
ra—1 1
ra—1
1
ra—1
0 0 0
0 1
n— 1 1
ra—1
1
ra—1
0 1
ra—1 1
ra—1
1
n— 1
1
ra—1
1
ra—1
1 1
га—1 n—1 1 1 га—1 ra—1 1
ra—1
0
1 1
n—1 ra—1
... 0
... 0
0 0
0 0
Characteristic equation is
A +
1
n1
n-2
A
n1
n2
A2
n1
A1
л 2 2n — 3 л n — 3 A2--—A +
n1
n1
spectrum is Specg (Sn) =
(2ra-3)+V4ra-3 l+V4ra2-8ra+5 (2ra-3)~V4ra-3 1 -1 l-V4ra2-8ra+5
2(n-l) 2(n-l) 2(n—1) ra-l ra-l 2(ra-l)
1 1 1 n- 2 n-2 1
Therefore, (SJJ)
lD/oO\ _ (4ra-7)+V4ra2-8ra+5
n— 1
>
Theorem 4.4. Tiie minimum dominating Randic energy of complete bipartite graph Kn,nof order 2n with vertex set {u1, u2,..., un, v1, v2,..., vn} is
(Kn,n) =
2л/птг1
n
+ 2.
< Let Kn,n be the complete bipartite graph of order 2n with vertex set {u1, u2,..., un, V1, v2,..., vn}. The minimum dominating set = D = {u1, v1}. The minimum
0
0
1
1
0
dominating Randic matrix is
RD (Kn,n) =
1 0 0 0 . J_ n 1 n 1 n n
0 0 0 0 . J_ n 1 n 1 n 1 n
0 0 0 0 . J_ n 1 n I n 1 n
0 0 0 0 . J_ n 1 n I n 1 n
1 ri 1 n 1 n J_ n .1 0 0 0
1 n 1 n I n J_ n .0 0 0 0
1 n 1 n I n J_ n .0 0 0 0
1 -П 1 n 1 n J_ n .0 0 0 0
Characteristic equation is A2n-4
D
A2
n — 1
n
Hence, spectrum is SpecD (Kn,n) = Therefore, RED(Kn,n) = + 2. >
A2 - 2A +
1 +
n1
n
Definition 4.1. The friendship graph, denoted by F3(n), is the graph obtained by taking n copies of the cycle graph C3 with a vertex in common. V(Fn) = 2n +1.
Теорема 4.5. The minimum dominating Randic energy of Friendship graph Fn is
REd (Fn) = n + 1.
(n)
< Let F3 be the friendship graph with 2n + 1 vertices. Here v1 be the center. The minimum dominating set = D = |vi}. The minimum dominating Randic matrix is
1
1
0
2n — 4
y/n-î л/п 1
Rd (F3 ) =
1 1 1 1
2 v^
1
2 v^ 1
1
0
1
2
0 0
0 0
Characteristic equation is
a( A +
Hence, spectrum is
1
1
2
0 0 0
A
1
0 0
0
1
2
0 0
n— 1
1
0
0
1
2
1
0 0 0 0
1 ~i
0 0 0 0
A--]=0.
SpecR (Fn ) Therefore, RED(F%) = n + 1. >
1 0
2 u 2
n
3 2
1 n 1 1
0
1
1
1
0
0
ra
1
1
2
2
Теорема 4.6. The minimum dominating Randic energy of Cocktail party graph Knx2 is
4 n- 6
RED (Knx 2) =
n — 1
< Let Knx2 be a Cocktail party graph of order 2n with vertex set {ui, u2,... ,un,vi,v2,... ,vn}. The minimum domina ting set = D = {ul, vl}. The minimum dominating minimum dominating Randic matrix is
RD (Knx2) =
1 1 1 1 ■0 1 1 1
2 n- 2 2n—2 2 n- 2 ■ 2 n- 2 2n—2 2 n- 2
1 0 1 1 1 0 1 1
2 n- 2 2ra-2 2 n- 2 ■ 2 n- 2 2ra-2 2 n- 2
1 1 0 1 1 1 0 1
2 n- 2 2 n- 2 2 n- 2 ■ 2 n- 2 2 n- 2 2 n- 2
1 1 1 0 1 1 1 0
2 n- 2 2 n- 2 2ra-2 2 n- 2 2 n- 2 2ra-2
0 1 1 1 ■1 1 1 1
2 n- 2 2ra-2 2 n- 2 ■ 2 n- 2 2ra-2 2 n- 2
1 0 1 1 1 0 1 1
2 n- 2 2n—2 2 n- 2 ■ 2 n- 2 2n—2 2 n- 2
1 1 0 1 1 1 0 1
2 n- 2 2 n- 2 2 n- 2 ■ 2 n- 2 2 n- 2 2 n- 2
1 1 1 0 1 1 1 0
2 n- 2 2 n- 2 2ra-2 2 n- 2 2 n- 2 2ra-2
Characteristic equation is
X
n— 1
X +
1
n1
n2
(X -1)
л 2 2n — 3 л n — 3 A2--—A +
n1
n1
Hence, spectrum is
SpecD (Knx2) =
2ra-3+V4ra-3 i 2ra-3~V4ra-3 n 2(n-l) 1 2(n-l) U
-1 n— 1
n — 1 n — 2
Therefore, REv(Knx2) = . >
Acknowledgement. The authors are thankful to the anonymous referee for valuable suggestions and comments for the improvement of the paper. Also, the first author is grateful to Dr. M. N. Channabasappa, Director and Dr. Shivakumaraiah, Principal, Siddaganga Institute of Technology, Tumkur, for their constant support and encouragement.
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1
References
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Received, August 6, 2016.
P. Siva Кота Reddy, Prof.
Department of Mathematics
Siddaganga Institute of Technology,
В. H. Road, Tumkuru-572 103, Karnataka, INDIA
E-mail: pskreddyOsit. ac. in; [email protected]
K. N. Prakasha, Prof. Department of Mathematics Vidyavardhaka College of Engineering
P. B. No. 206, Gokulam III Stage, Mysore-570 002, Karnataka, INDIA E-mail: prakashamathsOgmail. com
v. m. slddalingaswamy, Prof.
Department of Mathematics
JSS Academy of Technical Education
Uttarahalli-Kengeri Main Road, Bangalore-560 060, INDIA
E-mail: [email protected]
МИНИМАЛЬНАЯ ДОМИНИРУЮЩАЯ ЭНЕРГИЯ РАНДИЧА ГРАФА
Сива Кота Редди П., Пракаша К. Н., Сиддалингасвами В. М.
В данной работе мы ввели понятие и вычислили минимальную доминирующую энергию Рандича графа. Кроме того, были найдены верхняя и нижняя границы для минимальной доминирующей энергии Рандича.
Ключевые слова: минимальный доминирующий набор, минимальные доминирующие собственные значения Рандича, минимальная доминирующая энергия Рандича.
