Electronic Journal of Graph Theory and Applications 12 .
, 25Ae34 On D-distance .
magic labelings of shadow graph of some graphs Anak Agung Gede Nguraha .
Nur Inayahb .
Mohamad I.
Mustib a Department of Civil Engineering.
Universitas Merdeka Malang.
Jalan Terusan Raya Dieng 62 Ae 64 Malang.
Indonesia b Department of Mathematics.
Faculty of Science and Technology.
State Islamic University Syarif Hidayatullah.
Jl.
Ir H.
Juanda No.
95 Tangerang Selatan 15412.
Indonesia.
ngurah@unmer.
id, .
inayah, mohamad.
@uinjkt.
Corresponding author: nur.
inayah@uinjkt.
id (Nur Inaya.
Abstract Let G be a graph with vertex set V (G) and diameter diam(G).
Let D OI .
, 1, 2, 3, .
, diam(G)} PI : V (G) Ie .
, 2, 3, .
, |V (G)|} be a bijection.
The graph G is called D-distance magic, if sOOND .
is a constant for any vertex t OO V (G).
The graph G is called (, )-D-distance antimagic, if { sOOND .
: t OO V (G)} is a set {, , 2, .
, (|V (G)| Oe .
this paper, we study D-distance .
magic labelings of shadow graphs for D = .
, .
, .
, .
, and .
, .
Keywords: D-distance .
magic labeling.
D-distance .
magic graph, shadow graph Mathematics Subject Classification : 05C78 DOI: 10.
5614/ejgta.
Introduction We follow the terminologies and notations introduced in .
, 12, .
Let G be a simple graph with vertex set V (G) and diameter diam(G).
For two vertices s, t OO V (G), the distance between s and t is denoted by d.
, .
Let D be a set of distances in .
, 1, 2, 3, .
, diam(G)}, and I : V (G) Ie .
, 2, 3, .
, |V (G)|} be a bijection.
The neighborhood P of a vertex t OO V (G) under D is ND .
= .
OO V (G) : d.
, .
OO D}, and its weight is wD .
= sOOND .
If D = .
Received: 15 May 2023.
Revised: 24 October 2023.
Accepted: 26 October 2023.
On D-distance .
magic labelings of shadow graph of some graphs Ngurah et al.
= N .
= .
OO V (G) : st OO E(G)} and w.
= w.
= sOON .
If D = .
, .
= .
O N .
and w.
= I.
In the two next definitions, in case the graph G is a disconnected graph, diam(G) is the maximum diameter of its components.
Definition 1.
A bijection I : V (G) P Ie .
, 2, .
, |V (G)|} is called a D-distance magic (DM) labeling of a graph G, if wD .
= sOOND .
is a constant k for every vertex t OO V (G).
graph which admits a D-DM labeling is called a D-DM graph The constant k is called vertex sum of the labeling I.
If D = .
, a .
-DM labeling and a .
-DM graph are called a DM labeling and a DM graph, respectively .
These notions were independently introduced in .
, .
Definition 2.
Let I : V (G) Ie .
, 2, .
, |V (G)|} be a bijection.
If wD .
= wD .
for every s, t OO V (G), then I is called D-distance antimagic (DA) labeling of G and G is called a D-DA graph.
If .
D .
: t OO V (G)} is {, , 2, .
, (|V (G)| Oe .
}, where Ou 0 and > 0 are fixed integers, then I is called an (, )-D-DA labeling of G, and G is called an (, )-D-DA If D = .
, a .
-DA labeling .
-DA grap.
is called a DA labeling .
a DA grap.
If D = .
, an (, )-.
-DA labeling .
an (, )-.
-DA grap.
is called an (, )-DA labeling .
an (, )-DA grap.
Many results on these subjects have been published.
Some results on D-DM labeling can be seen in .
, 4, 12, 13, 14, .
, results on D-DA labeling can be seen in .
, 3, 5, 8, 13, .
, recent results on .
, .
-DM labeling on shadow graph of some graphs can be seen in .
, and the complete results can be seen in .
Let G be a graph with no isolated vertices.
The shadow graph of a graph G, denoted by D2 (G), is the graph constructed from 2G by joining each vertex in the second component to the neighbors of the corresponding vertex in the first component.
We denote the first component by G with vertex set V (G) = .
i : 1 O i O |V (G)|} and the second one by GA with the corresponding vertex set V (GA ) = .
Ai : 1 O i O |V (GA )|}.
From the definition of D2 (G), every vertex u and uA has the same neighbors, namely N .
= N .
A ), in D2 (G), and d.
, uA ) = 2.
Examples of shadow graphs of P4 and C4 are given in Figures 1 .
and 1 .
, respectively.
In this paper, we give some necessary conditions for D2 (G) to be D-DM as well as D-DA, where G is a regular graph.
Also, we prove the existence and nonexistence of the D-DM labeling and the (, )-D-DA labeling of shadow graph of cycles and complete bipartite graphs for D = .
, .
, .
, .
, and .
, .
Main Results Our first result shows the relationship between a D-DM graph and an (, .
-DA -DA graph for some D.
DA OO .
, 2, 3, .
, diam(G)}.
On D-distance .
magic labelings of shadow graph of some graphs Ngurah et al.
Figure 1.
The graphs D2 (P4 ) and D2 (C4 ).
Lemma 2.
Let G be a graph with p vertices and diameter diam(G).
Let DO OI .
, 2, 3, .
diam(G)} and D = DO O .
If G is a DO -DM graph with vertex sum k, then G is a .
1, .
-D-DA graph.
If G is a D-DM graph with vertex sum k, then G is a .
Oe p, .
-DO -DA graph.
Proof.
Let I be a DO -DM labeling of G with vertex sum k.
Then wDO .
= sOONDO .
= k for every t OO V (G).
Now, .
D .
: t OO V (G)} = {I.
sOONDO .
: t OO V (G)} = {I.
k : t OO V (G)}.
Since I.
OO .
, 2, 3, .
, .
, then .
D .
: t OO V (G)} = .
1, k 2, k 3, .
, k .
P Let I be a D-DM labeling of G with vertex sum k.
Then .
DO .
: t OO V (G)} = { sOOND .
Oe I.
: t OO V (G)} = .
Oe I.
: t OO V (G)} = .
Oe p, k Oe p 1, k Oe p 2, .
, k Oe .
The next results show that the graph D2 (G) has no a D-DA labeling as well as a D-DM labeling for some D.
Lemma 2.
Let G a graph with no isolated vertices.
The graph D2 (G) is not a DA graph and it is not a .
, .
-DM graph.
The graph D2 (G) is not a .
, .
-DA graph and it is not a .
-DM graph.
Proof.
Assume that D2 (G) is a DA graph P with a DA labeling PI.
Let us consider vertices A dan uA .
Since N .
= N .
A ), then w.
= I.
vOON .
vOON .
A ) A contradiction to the fact that w.
= w.
with a .
, .
-DM labeling I.
Then I.
P Next, suppose that D2 (G) is an .
-DM graph vOON .
vOON .
A ) I.
Since N .
= N .
), then I.
= I.
A ).
It is a contradiction, since I is a bijection.
Notice that N.
= N.
A ) = .
, uA } O .
OO V (G) : d.
, .
= .
O .
A OO V (GA ) :
A A
, v ) = .
= .
A } O .
OO V (G) : d.
, .
= .
O .
A OO V (GA ) : d.
A , v A ) = .
, and N.
A ) = .
O .
OO V (G) : d.
, .
= .
O .
A OO V (GA ) : d.
A , v A ) = .
By similar argument as in the first part, we can show that D2 (G) is not .
, .
-DA and it is not .
-DM.
The following results provide some necessary conditions for D2 (G) to be a D-DM graph or an (, )-D-DA graph for some D.
On D-distance .
magic labelings of shadow graph of some graphs Ngurah et al.
Lemma 2.
Let G be a graph with p vertices, |N .
| = r1 , and |N.
| = r2 for each u OO V (G).
If D2 (G) is a DM graph, then its vertex sum is k = r1 .
If D2 (G) is a .
, .
-DM graph, then its vertex sum is k = .
Proof.
The graph D2 (G) has 2p vertices and |N .
| = 2r1 for each u OO V (D2 (G)).
If D2 (G) is DM with vertex sum is k, then 2pk = 2r1 .
2 3 A A A 2.
= 2pr1 .
For each u OO V (D2 (G)), |N.
| = |.
, uA }| |.
OO V (G) : d.
, .
= .
| |.
A OO V (GA ) : d.
A , v A ) = .
| = 2r2 2.
So.
If D2 (G) is .
, .
-DM with vertex sum is k, then 2pk = .
2 3 A A A 2.
= 2p.
Theorem 2.
Let G be a graph with p vertices, |N .
| = r1 , and |N.
| = r2 for each u OO V (G).
If D2 (G) is an .
, 1 )-.
, .
-DA graph, then 1 is odd and for r1 << p, 1 O 2r1 Oe 1.
If D2 (G) is an .
, 2 )-.
-DA graph, then 2 is odd and for r2 << p, 2 O 2r2 Oe 1.
Proof.
Notice that, for every u OO V (D2 (G)), |N.
| = |.
O N .
| = 2r1 1.
Next, let D2 (G) be an .
, 1 )-.
, .
DA graph.
Then .
: u OO V (D2 (G))} = .
, 1 1 , 1 21 , .
, 1 .
p Oe .
1 }.
The sum of all vertex weights is 1 .
1 ) .
21 ) A A A .
p Oe .
1 ) = 2p1 1 p.
p Oe .
This sum contains 2r1 1 times each vertex label, since |N.
| = 2r1 1 for every u OO V (D2 (G)).
So, 2p1 1 p.
p Oe .
= .
2 A A A 2.
= .
21 1 .
p Oe .
= .
Since .
is an odd integer and 21 is an even integer, then 1 .
p Oe .
must be an odd integer.
Hence, 1 is an odd integer.
Next, the minimum possible .
weight is 1 2 3 A A A .
and its maximum is 2p .
p Oe .
p Oe .
p Oe .
A A A .
p Oe 2r1 ).
Hence, 1 Ou .
and 1 .
p Oe .
1 O .
p Oe r1 ).
So, 1 O 2r1 1 Oe 2p Oe 1 For a small r1 and a large p, then 0 < 2r12pOe1 < 1.
Hence, 1 O 2r1 Oe 1, since 1 is an odd For every u OO V (D2 (G)), |N.
| = |.
A } O .
OO V (G) : d.
, .
= .
O .
A OO V (GA ) :
A , v A ) = .
| = 2r2 1.
By the same argument as in the first part, we have the desire results.
Lemma 2.
Let G be a graph with p vertices and d be a positive integer.
If I1 is an .
, 1 )-.
, .
-DA labeling of D2 (G), then |I1 .
Oe I1 .
A )| = d1 for every pair u and uA in V (D2 (G)).
If I2 is an .
, 2 )-.
-DA labeling of D2 (G), then |I2 .
Oe I2 .
A )| = d2 for every pair u and uA in V (D2 (G)).
On D-distance .
magic labelings of shadow graph of some graphs Ngurah et al.
Proof.
For every pair u and uA in V (D2 (G)), w.
= I1 .
vOON .
I1 .
= 1 d1 1 and w.
A ) = I1 .
A ) vOON .
A ) I1 .
= 1 d2 1 for some d1 , d2 OO .
, 1, 2, 3, .
, 2p Oe .
Since vOON .
I1 .
= vOON .
A ) I1 .
, then I1 .
Oe I1 .
A ) = .
1 Oe d2 )1 = d1 or I1 .
A ) Oe I1 .
= .
2 Oe d1 )1 = Oed1 .
For every pair uP and uA in V (D2 (G)), w.
= I2 .
A ) vOOSOS A I2 .
= 2 d3 2 and w.
A ) = I2 .
vA OOSOS A I2 .
A ) = 2 d4 2 for some d3 , d4 OO .
, 1, 2, 3, .
, 2p Oe .
, where A A A A A : d.
, .
= .
and S = .
OO V (G ) : d.
, v ) = .
Since, vOOSOS A I2 .
= P= .
OO V (G) A then |I2 .
A ) Oe I2 .
| = |.
3 Oe d4 ).
v OOSOS Next, we consider m copies of the graph D2 (G), namely mD2 (G), where G = Cn and Kn,n .
Notice that mD2 (G) O = D2 .
G).
By Lemma 2.
2, the graphs mD2 (Cn ) and mD2 (Kn,n ) are not DA and .
, .
-DA.
Also, they are not .
, .
-DM and .
-DM.
In the next theorem, we show that mD2 (Cn ) has DM and .
, .
-DM labelings for every integer m Ou 1 and n Ou 3.
Theorem 2.
For every integer m Ou 1 and n Ou 3, the graph mD2 (Cn ) is DM and .
, .
-DM.
Proof.
Let V .
D2 (Cn )) = .
i,j , uAi,j : 1 O i O n, 1 O j O .
and E.
D2 (Cn )) = .
i,j ui 1,j , uAi,j uAi 1,j , uAi,j ui 1,j , uAi 1,j ui,j , : 1 O i O n Oe 1, 1 O j O .
O .
n,j u1,j , uAn,j uA1,j , uAn,j u1,j , uA1,j un,j :
1 O j O .
For 1 O j O m, let Aj = {.
i,j , uAi,j } : 1 O i O .
and Bj = {{.
Oe .
n i, 2nm 1 Oe .
Oe .
n Oe .
: 1 O i O .
It is clear that for k = l.
Ak O Al = OI and Bk O Bl = OI.
Also.
A = Om j=1 Aj = V (D2 (Cn )) and B = Oj=1 Bj = .
, 2, 3, .
, 2n.
Hence.
A is a partion of .
, 2, 3, .
, 2m.
and B is a partition of V (D2 (Cn )).
Next, one can check that, for 1 O j O m, and any bijection I : A Ie B we have w.
1,j ) = w.
A1,j ) = I.
n,j ) I.
An,j ) I.
2,j ) I.
A2,j ) = 4mn 2, w.
i,j ) = w.
Ai,j ) = I.
iOe1,j ) I.
AiOe1,j ) I.
i 1,j ) I.
Ai 1,j ) = 4mn 2 for 1 O i O n Oe 1, and w.
n,j ) = w.
An,j ) = I.
nOe1,j ) I.
AnOe1,j ) I.
1,j ) I.
A1,j ) = 4mn 2.
Therefore.
I is a DM labeling of mD2 (Cn ) with vertex sum 4mn 2.
Now, we show that I : A Ie B is also a .
, .
-DM labeling of mD2 (Cn ).
To do this, we consider three the following cases:
Case n = 3.
For 1 O j O m and 1 O i O 3, w.
i,j ) = w.
Ai,j ) = I.
i,j ) I.
Ai,j ) = 6m 1.
Thus.
I is a .
, .
-DM labeling of mD2 (C3 ) with vertex sum 6m 1.
Case n = 4.
For 1 O j O m, w.
1,j ) = w.
A1,j ) = w.
3,j ) = w.
A3,j ) = I.
1,j ) I.
A1,j ) I.
3,j ) I.
A3,j ) = 16m 2, and w.
2,j ) = w.
A2,j ) = w.
4,j ) = w.
A4,j ) = I.
2,j ) I.
A2,j ) I.
4,j ) I.
A4,j ) = 16m 2.
Thus.
I is a .
, .
-DM labeling of mD2 (C4 ) with vertex sum 16m 2.
Case n Ou 5.
For 1 O j O m, w.
1,j ) = w.
A1,j ) = I.
1,j ) I.
A1,j ) I.
3,j ) I.
A3,j ) I.
nOe1,j ) I.
AnOe1,j ) = 6mn 3, w.
2,j ) = w.
A2,j ) = I.
2,j ) I.
A2,j ) I.
4,j ) I.
A4,j ) I.
n,j ) I.
An,j ) = 6mn 3, w.
i,j ) = w.
Ai,j ) = I.
i,j ) I.
Ai,j ) I.
i 2,j ) I.
Ai 2,j ) I.
iOe2,j ) I.
AiOe2,j ) = 6mn 3 for 3 O i O n Oe 2, w.
nOe1,j ) = w.
AnOe1,j ) = I.
nOe1,j ) I.
AnOe1,j ) I.
1,j ) I.
A1,j ) I.
nOe3,j ) I.
AnOe3,j ) = 6mn 3, and w.
n,j ) = w.
An,j ) = I.
n,j ) I.
An,j ) I.
2,j ) I.
A2,j ) I.
nOe2,j ) I.
AnOe2,j ) = 6mn 3.
Thus.
I is a .
, .
-DM labeling of mD2 (Cn ), n Ou 5, with vertex sum 6mn 3.
As an example, let consider the case m = 1.
In this case, we redefine vertex and edge sets of D2 (Cn ) as follows: V (D2 (Cn )) = .
i , uAi : 1 O i O .
and E(D2 (Cn )) = .
i ui 1 , uAi uAi 1 :
On D-distance .
magic labelings of shadow graph of some graphs Ngurah et al.
1 O i O n Oe .
O .
n u1 , uAn uA1 } O .
Ai ui 1 , uAi 1 ui : 1 O i O n Oe .
O .
An u1 , uA1 un }.
Also.
A = {.
i , uAi } : 1 O i O .
and B = {.
, 2n 1 Oe .
: 1 O i O .
Next, let I(.
i , uAi }) = .
, 2n 1Oe.
for 1 O i O n.
Then w.
1 ) = w.
A1 ) = [I.
n ) I.
An )] [I.
2 ) I.
A2 )] = .
2n Oe .
= 4n 2, w.
i ) = w.
Ai ) = [I.
iOe1 ) I.
AiOe1 )] [I.
i 1 ) I.
Ai 1 )] = .
Oe 1 2n 1 Oe i .
1 2n 1 Oe i Oe .
= 4n 2 for 1 O i O n Oe 1, and w.
n ) = w.
An ) = [I.
nOe1 ) I.
AnOe1 )] [I.
1 ) I.
A1 )] = .
Oe1 n .
= 4n 2.
Hence.
I is a DM labeling of D2 (Cn ) with vertex sum 4n 2.
Next, we show that I is also a .
, .
-DM labeling D2 (Cn ).
Case n = 3.
i ) = w.
Ai ) = I.
i ) I.
Ai ) = 7 for 1 O i O 3.
Case n = 4.
1 ) = w.
A1 ) = w.
3 ) = w.
A3 ) = I.
1 ) I.
A1 ) I.
3 ) I.
A3 ) = 18 and w.
2 ) = w.
A2 ) = w.
4 ) = w.
A4 ) = I.
2 ) I.
A2 ) I.
4 ) I.
A4 ) = 18.
Case n Ou 5.
1 ) = w.
A1 ) = I.
1 ) I.
A1 ) I.
3 ) I.
A3 ) I.
nOe1 ) I.
AnOe1 ) = 6n 3, w.
2 ) = w.
A2 ) = I.
2 ) I.
A2 ) I.
4 ) I.
A4 ) I.
n ) I.
An ) = 6n 3, w.
i ) = w.
Ai ) = I.
i ) I.
Ai ) I.
i 2 ) I.
Ai 2 ) I.
iOe2 ) I.
AiOe2 ) = 6n 3 for 3 O i O n Oe 2, w.
nOe1 ) = w.
AnOe1 ) = I.
nOe1 ) I.
AnOe1 ) I.
1 ) I.
A1 ) I.
nOe3 ) I.
AnOe3 ) = 6n 3, and w.
n ) = w.
An ) = I.
n ) I.
An ) I.
2 ) I.
A2 ) I.
nOe2 ) I.
AnOe2 ) = 6n 3.
Hence.
I is a .
, .
-DM labeling of D2 (C3 ).
D2 (C4 ), and D2 (Cn ), n Ou 5, with vertex sum 7, 18 , and 6n 3, respectively.
Next, we consider the (, )-.
, .
-DA and (, )-.
-DA labelings of the graph mD2 (Cn ).
Lemma 2.
Let m Ou 1 and n Ou 3 be integers.
If the graph mD2 (Cn ) is .
, 1 )-.
, .
-DA, then 1 = 1, 1 = 4nm 3 and 1 = 3, 1 = 2nm 4.
If the graph mD2 (Cn ) is .
, 2 )-.
-DA graphs, then 2 = 1, 2 = 4nm 3 and 2 = 3, 2 = 2nm 4.
Proof.
By Theorem 2.
1 and equation .
, we have the desire results.
As a consequence of Lemma 2.
1 and Theorem 2.
2, we have the following result.
Corollary 2.
For every integer m Ou 1 and n Ou 3, the graph mD2 (Cn ) is .
nm 3, .
, .
DA.
For every integer m Ou 1, the graph mD2 (C3 ) is .
, .
-DA.
For every integer m Ou 1, the graph mD2 (C4 ) is .
m 2, .
-DA.
For every integer m Ou 1 and n Ou 5, the graph mD2 (Cn ) is .
mn 3, .
-DA.
Lemma 2.
If mn O 1, 2 .
, then the graph mD2 (Cn ) is not .
, .
, .
-DA and it is not .
, .
-DA for some integer 1 and 2 .
Proof.
Due to Lemma 2.
4, if I is an (, .
, .
(, .
)-DA labeling of mD2 (Cn ), then |I.
Oe I.
A )| = 3d for some positive integer d and for every pair u, uA OO V .
D2 (Cn )).
Hence, 2nm O 0 .
or nm O 0 .
On D-distance .
magic labelings of shadow graph of some graphs Ngurah et al.
Next, let us consider the graph D2 (Cn ), where n O 0 .
It is not easy for us to prove whether D2 (Cn ) is .
n 4, .
, .
-DA or not.
We only have the following results.
By equation .
, the graph D2 (C3 ) is not .
, .
, .
-DA.
Let D2 (C6 ) is .
, .
, .
-DA, then .
u OO V (D2 (C6 ))} = .
, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, .
Since there is a unique way to express 16 and 49 as a sum of five numbers in the set .
, 2, 3, .
, .
, that is 16 = 1 2 3 4 6 and 49 = 7 9 10 11 12, and due to Lemma 2.
4, we have two possibilities to label of vertices of D2 (C6 ) as in the Figure 2.
We can verify that the labelings do not lead to a .
, .
, .
DA labeling of D2 (C6 ).
So, the graph D2 (C6 ) is not .
, .
, .
-DA.
By the same arguments.
D2 (C3 ) is not .
, .
-DA and D2 (C6 ) is not .
, .
-DA.
Figure 2.
The possibilities to label of D2 (C6 ) Problem 1.
Decide if there exists a .
n 4, .
, .
n 4, .
Oe .
)-DA labeling of D2 (Cn ) for every integer 9 O n O 0 .
Next, we consider the shadow graph mD2 (Kn,n ).
In the next result, we show that mD2 (Kn,n ) is a DM graph as well as a .
, .
-DM graph.
Theorem 2.
For every integer m, n Ou 1, the graph G = mD2 (Kn,n ) is DM and .
, .
-DM.
Proof.
For 1 O j O m, let V (G) = V1,j O V2,j O V1,j O V2,j and E(G) = V1,j V2,j O V1,j V2,j O V2,j V1,j O V1,j V2,j , where V1,j = .
i,j : 1 O i O .
V2,j = .
i,j : 1 O i O .
V1,j = .
Ai,j :
1 O i O .
V2,j = .
i,j : 1 O i O .
, and Vi,j Vk,l means that every vertex in Vi,j is adjacent to each vertex in Vk,l and vice versa.
Next, for 1 O j O m, let S1,j = {.
Oe .
n i : 1 O i O .
S2,j = .
Oe .
n i : 1 O i O .
S3,j = .
mn 1 Oe .
Oe .
n Oe i : 1 O i O .
, and S4,j = .
mn 1Oe.
Oe.
nOei : 1 O i O .
It is clear that.
P for 1 O j O m.
S1,j OS2,j OS3,j OS4,j = .
, 2, 3, .
, 4m.
, sOOS1,j s = 21 n.
Oe .
, sOOS2,j s = 21 n.
mn 2n.
Oe .
, sOOS3,j s = 2 n.
mn Oe 2n.
Oe .
Oe n .
, and sOOS4,j s = 2 n.
mn Oe 2n.
Oe .
Oe n .
Next, for 1 O j O m, label each vertex in V1,j by every number in S1,j , each vertex in A V2,j by every number in S2,j , each vertex in V1,j by every number in S4,j , and each vertex in A V2,j by every number in S3,j .
Then, for 1 O i O n and 1 O j O m, w.
i,j ) = w.
Ai,j ) = On D-distance .
magic labelings of shadow graph of some graphs Ngurah et al.
mn 2n.
Oe .
12 n.
mn Oe 2n.
Oe .
Oe n .
= n.
, and w.
i,j ) = A ) = 21 n.
Oe .
12 n.
mn Oe 2n.
Oe .
Oe n .
= n.
Thus, mD2 (Kn,n ) w.
i,j is a DM graph.
Next, we show that the labeling is also a .
, .
-DM labeling of mD2 (Kn,n ).
For 1 O i O n and 1 O j O m, w.
i,j ) = w.
Ai,j ) = 12 n.
Oe.
21 n.
mnOe2n.
Oe.
Oen .
= n.
A and w.
i,j ) = w.
i,j ) = 12 n.
mn 2n.
Oe.
12 n.
mnOe2n.
Oe.
Oen .
= n.
Hence, mD2 (Kn,n ) is a .
, .
-DM graph.
Next, we provide an illustration of the proof of Theorem 2.
3 for m = 1.
First, redefine vertex and edge sets of D2 (Kn,n ) as follows: V (D2 (Kn,n )) = V1 O V2 O V1A O V2A and E(D2 (Kn,n )) = V1 V2 O V1A V2A O V1A V2 O V2A V1 , where V1 = .
i : 1 O i O .
V2 = .
i : 1 O i O .
V1A = .
Ai : 1 O i O .
V2A = .
iA : 1 O i O .
Also.
S1 = .
, 2, 3, .
, .
S2 = .
1, n 2, n 3, .
, 2.
Oe 1, 3n Oe , 3n .
Obviously, 4 = .
n, 4n Oe 1, 4n Oe 2, .
P3 = .
n, 3n P2, .
, 2n 1 .
, and SP sOOS1 s = 2 n.
, sOOS2 s = 2 n.
, sOOS3 s = 2 n.
, and sOOS4 s = 2 n.
Finally, label every vertex in V1 by each member of S1 , every vertex in V2 by each member of S2 , every vertex in V1A by each member of S4 , and every vertex in V2A by each member of S3 .
Under this labeling, for 1 O i O n, w.
i ) = w.
Ai ) = 12 n.
12 n.
= n.
, and w.
i ) = w.
iA ) = 12 n.
12 n.
= n.
So.
D2 (Kn,n ) is a DM graph.
Next, for 1 O i O n, w.
i ) = w.
Ai ) = 21 n.
12 n.
= n.
, and w.
i ) = w.
iA ) = n.
21 n.
= n.
So.
D2 (Kn,n ) is a .
, .
-DM graph.
As a consequence of Lemma 2.
1 and Theorem 2.
3, we have the following result.
Corollary 2.
The graph mD2 (Kn,n ) is .
1, .
, .
-DA and .
mnOe4m .
, .
-DA for every integer m, n Ou 1.
By a similar argument as in the proof of Lemma 2.
6, we have the following lemma.
Lemma 2.
If mn O 1, 2 .
, then the graph mD2 (Kn,n ) is not .
, .
, .
-DA and it is not .
, .
-DA for some integer 1 and 2 .
The problem related to these results are as follows.
Problem 2.
Decide if there exists a (, .
, .
(, .
)-DA labeling of (D2 (Kn,n )) for every integer n O 0 .
Problem 3.
For every integer m Ou 1 and n1 = n2 Ou 1, find a DM labeling and an (, )-.
, .
DA labeling of the graph mD2 (Kn1 ,n2 ).
Conclusion In this paper, we study D-DM labeling and (, )-D-DA labeling of shadow graphs for D OO {.
, .
, .
, .
, .
, .
We provide some necessary conditions for the shadow graph of a regular graph to be D-DM or (, )-{D}-DA.
Also, we prove the existence and nonexistence of D-DM labeling and (, )-{D}-DA labeling for the graphs mD2 (Cn ) and mD2 (Kn,n ).
Our results also give an example if D2 (G) is D-DM.
G needs not to be D-DM.
Namely.
D2 (Cn ) is DM for every n Ou 3, however.
Cn is not DM for n = 4.
On D-distance .
magic labelings of shadow graph of some graphs Ngurah et al.
References