The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5)

He Shuping; Wu Qiong

doi:doi:10.11648/j.ijssam.20251001.12

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The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph D_n(1,5)

He Shuping

, Wu Qiong^*

Published in International Journal of Systems Science and Applied Mathematics (Volume 10, Issue 1)

Received: 29 December 2024 Accepted: 14 January 2025 Published: 10 February 2025

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Abstract

For j ≤ k, the L(j, k)- labeling problem arose form the code assignment problem of the wireless network. That is, let n,j,k be non-negative real numbers with j ≤ k, an n-L(j, k)-labeling of a graph G is mapping f: V(G)→[0, n] such that |f(u)-f(v)| ≥ j if d(u, v)=1, and |f(u)-f(v)|≥k if t d(u, v)=2. The span of f is the difference between the maximum and minimum labeling numbers assigned by f. The L(j, k)-labeling number of graph G, denoted by λ_(j,k) (G), is the minimum span of all L(j, k)-labeling of G. The infinite distance graph, denoted by D (d₁,d₂,…,d_k), has the set Z of integers as a vertex set and in which two vertices i,j∈Z are adjacent if and only if |i-j|∈D. The finite distance graph, denoted by D_n (d₁,d₂,…,d_k), is the subgraph of D(d₁,d₂,…,d_k) induced by vertices {0,1,…,n-1}. This paper determines the L(j, k)-labeling number and the circular L(j, k)-labeling number of distance graph D_n (1,5) for 2j ≤ k.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 10, Issue 1)
DOI	10.11648/j.ijssam.20251001.12
Page(s)	7-11
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

The L(j, k)-labeling Number, The Circular L(j, k)-labeling Number, Code Assignment, Distance Graph

1. Introduction

For

j \leq k

, the

L (j, k)

-labeling, first introduced by Yeh

[1]

, arose from the code assignment of wireless network proposed by Bertossi and Bonuccelli

[2]

. A vertex and its label will represent a station and its corresponding code, respectively. If two stations can transmit to each other directly, their corresponding vertices are connected by an edge, for convenience, these two stations are called adjacent stations, and the two stations are at distance two if they are adjacent to a common station. To avoid the interferences in the wireless network, labels of vertices of a graph should satisfy some conditions. For example, a Packet Radio Network using CDMA has two types of interference: Direct interference and Secondary interference. Direct interference is caused by the adjacent stations, and stations at distance two cause the Secondary interference and direct interference simultaneously. In order to avoid these two types of interferences, the adjacent vertices are assigned labels whose difference is at least

j

and vertices at distance two are assigned labels whose difference is at least

k

, where

j \leq k

Let

n, j, k

be non-negative real numbers with

j \leq k

, an

n

L (j, k)

-labeling of a graph

G

is mapping

f

V (G) \to [0, n]

such that

|f (u) - f (v)| \geq j

d (u

v) = 1

, and

|f (u) - f (v)| \geq k

if t

d (u

v) = 2

. The span of

f

is the difference between the maximum and minimum labels assigned by

f

. The

L (j, k)

-labeling number of graph

G

, denoted by

λ_{j, k} (G)

, is the minimum span of all

L (j, k)

-labelings of

G .

L (j, k)

-labeling numbers of some graphs for

j \leq k

have been introduced in some literatures. For instance, Wu, Shiu and Sun

[3]

investigated the

L (j, k)

-labeling number of cartesian product graph of a path and a cycle, Wu also studied on

L (j, k)

-labeling number of the strong product graph of path and cycle

[4]

, wheel graph and Cactus graph

[5, 6]

. Guo and Wu

[7]

determined the

L (j, k)

-labeling numbers of book graph. Rao et al. studied the

L (j, k)

-labeling numbers of cartesian product graph of arbitrary length of three paths with

k > j

in reference

[8]

Heuvel et al.

[9]

first introduced the circular

L (j, k)

-labeling of a graph. They used the “circular distance” to instead of the “linear distance”, where the linear distance of two point a and b on the line is defined as

| a - b |,

and the circular distance of two points a and b on the cycle with circumference

σ

is defined as

{|a - b|}_{σ} = \min \{|a - b|, σ - |a - b|\}

Let

σ

be non-negative real numbers, a circular

σ

L (j, k)

-labeling of a graph

G

is mapping

f

V (G) \to [0, σ)

such that

{|f (u) - f (v)|}_{σ} \geq j

d (u

v) = 1

, and

{|f (u) - f (v)|}_{σ} \geq k

d (u

v) = 2

. The circular

L (j, k)

-labeling number of G, denoted by

σ_{j, k} (G),

is the minimum

σ

such that there exists a circular

σ

L (j, k)

-labeling of G. The circular

L (j, k)

-labeling number of graphs have been studied in several articles, please refer to

[10-16]

Definition 1.1

[17]

Let

D = \{d_{1}, d_{2}, \dots, d_{k}\}

, where

d_{i} (i = 1, 2, \dots, k)

are positive integers such that

d_{1} < d_{2} < \dots < d_{k}

. The (infinite) distance graph

D (d_{1}, d_{2}, \dots, d_{k})

has the set

Z

of integers as a vertex set and in which two vertices

i, j \in Z

are adjacent if and only if

|i - j| \in D

. The finite distance graph

D_{n} (d_{1}, d_{2}, \dots, d_{k})

is the subgraph of

D (d_{1}, d_{2}, \dots, d_{k})

induced by vertices

\{0, 1, \dots, n - 1\}

Lemma 1.1

[3]

If graph

H

is an induced subgraph of graph

G

, then

λ_{j, k} (G) \geq λ_{j, k} (H)

Lemma 1.2

[3]

Let

j

and

k

be two positive numbers with

j \leq k

. Suppose

G

is a graph and

H

is an induced subgraph of

G

. Then

σ_{j, k} (G) \geq σ_{j, k} (H)

Lemma 1.3

[3]

Suppose

f

is an

L (j, k)

-labeling of graph

G

. Let

σ = \max \{\max_{d_{G} (u, v) = 2} \{k + |f (u) - f (v)|\}, \max_{d_{G} (u, v) = 1} \{j + |f (u) - f (v)|\}\}

If the image of

f

lies in

[0, σ)

, then

f

is a circular

σ

L (j, k)

-labeling of

G

2. Main Results

In this section, we mainly study on the

L (j, k)

-labeling numbers and the circular

L (j, k)

-labeling numbers of distance graph

D_{n} (1, 5)

for

n \geq 7

and

k \geq 2 j

. For

1 \leq n \leq 5

, the distance graph

D_{n} (1, 5)

is isomorphic to path

P_{n}

, and

D_{6} (1, 5)

is isomorphic to the cycle

C_{6}

. The

L (j, k)

-labeling numbers and the circular

L (j, k)

-labeling numbers of path and cycle have been discussed by Niu

[18]

and Wu, Shiu and Sun

[3]

, respectively, for

k \geq 2 j

. For convenience, let

G = D_{n} (1, 5)

Theorem 2.1 Let

n

be an integer and

j, k

be non-negative real numbers with

k \geq 2 j

. For

7 \leq n \leq 8

λ_{j, k} (G) = 3 k

Proof: Define a labeling function

f

for the graph

G

as follows.

f (i) = \{\begin{matrix} ik, 0 \leq i \leq 3, \\ {[i + 1]}_{4} k, 4 \leq i \leq n - 1 . \end{matrix}

As Figure 1 shows, it is not difficult to verify that

f

is an

L (j, k)

-labeling of graph

G

with

span (f) = 3 k

, where that is,

λ_{j, k} (G) \leq 3 k

Download: Download full-size image

Figure 1. The labeling

f

for distance graph

D_{8} (1, 5)

On the other hand, any pair of vertices

0

2

4

and

6

are at distance two, according to the definition of

L (j, k)

-labeling, it is at least

k

for the difference between any two labels of vertices

0

2

4

and

6

, thus,

λ_{j, k} (G) \geq 3 k

Hence,

λ_{j, k} (G) = 3 k

Theorem 2.2 Let

n

be an integer and

j, k

be non-negative real numbers with

k \geq 2 j

. For

n \geq 9

λ_{j, k} (G) = j + 3 k

Proof: Given a labeling

f

for graph

G

as follows.

f (i) = {[i]}_{2} j + {[⌊\frac{i}{2}⌋]}_{4} k

0 \leq i \leq n - 1

For example, the labeling

f

D_{9} (1, 5)

is shown in Figure 2.

Download: Download full-size image

Figure 2. The labeling

f

for distance graph

D_{9} (1, 5)

For any vertex

i

, according to the symmetry of the distance graph, it is needed to verify the labels of vertices

i + 1

and

i + 5

which are adjacent to vertex

i

, and the labels of vertices

i + 2, i + 4, i + 6

i + 10

which are 2 distance apart from vertex

i

1. The differences between

f (v_{i + 1})

and

f (v_{i})

f (v_{i + 5})

and

f (v_{i})

are shown as follows.

|f (v_{i + 1}) - f (v_{i})| = |[i + 1]_{2} j + {[⌊\frac{i + 1}{2}⌋]}_{4} k - [i]_{2} j - {[⌊\frac{i}{2}⌋]}_{4} k|

= |([i + 1]_{2} - [i]_{2}) j + ({[⌊\frac{i + 1}{2}⌋]}_{4} - {[⌊\frac{i}{2}⌋]}_{4}) k|

= \{\begin{matrix} j, & if i is even, \\ k - j or j + 3 k, & if i is odd, \end{matrix} \geq j .

|f (v_{i + 5}) - f (v_{i})| = |[i + 5]_{2} j + {[⌊\frac{i + 5}{2}⌋]}_{4} k - [i]_{2} j - {[⌊\frac{i}{2}⌋]}_{4} k|

= |([i + 5]_{2} - [i]_{2}) j + ({[⌊\frac{i + 5}{2}⌋]}_{4} - {[⌊\frac{i}{2}⌋]}_{4}) k|

= \{\begin{matrix} j + 2 k or 2 k - j, & if i is even, \\ j + k or 3 k - j, & if i is odd, \end{matrix} \geq j .

2. The differences between

f (v_{i + 2})

and

f (v_{i})

f (v_{i + 4})

and

f (v_{i})

f (v_{i + 6})

and

f (v_{i})

f (v_{i + 10})

and

f (v_{i})

are shown as follows.

|f (v_{i + 2}) - f (v_{i})| = |[i + 2]_{2} j + {[⌊\frac{i + 2}{2}⌋]}_{4} k - [i]_{2} j - {[⌊\frac{i}{2}⌋]}_{4} k|

= |([i + 2]_{2} - [i]_{2}) j + ({[⌊\frac{i + 2}{2}⌋]}_{4} - {[⌊\frac{i}{2}⌋]}_{4}) k|

= k or 3 k \geq k .

|f (v_{i + 4}) - f (v_{i})| = |[i + 4]_{2} j + {[⌊\frac{i + 4}{2}⌋]}_{4} k - [i]_{2} j - {[⌊\frac{i}{2}⌋]}_{4} k|

= |([i + 4]_{2} - [i]_{2}) j + ({[⌊\frac{i + 4}{2}⌋]}_{4} - {[⌊\frac{i}{2}⌋]}_{4}) k|

= 2 k \geq k .

|f (v_{i + 6}) - f (v_{i})| = |[i + 6]_{2} j + {[⌊\frac{i + 6}{2}⌋]}_{4} k - [i]_{2} j - {[⌊\frac{i}{2}⌋]}_{4} k|

= |([i + 6]_{2} - [i]_{2}) j + ({[⌊\frac{i + 6}{2}⌋]}_{4} - {[⌊\frac{i}{2}⌋]}_{4}) k|

= k or 3 k \geq k .

|f (v_{i + 10}) - f (v_{i})| = |[i + 10]_{2} j + {[⌊\frac{i + 10}{2}⌋]}_{4} k - [i]_{2} j - {[⌊\frac{i}{2}⌋]}_{4} k|

= |([i + 10]_{2} - [i]_{2}) j + ({[⌊\frac{i + 10}{2}⌋]}_{4} - {[⌊\frac{i}{2}⌋]}_{4}) k|

= k or 3 k \geq k .

According to the above verification,

f

satisfies the definition of

L (j, k)

-labeling, that is,

f

is an

L (j, k)

-labeling of distance graph

G

with

span (f) = j + 3 k

, thus,

λ_{j, k} (G) \leq j + 3 k

On the other hand, suppose

λ_{j, k} (G) = λ < j + 3 k

. Let

f

be an

L (j, k)

-labeling of distance graph

G

, and

I_{0} = [0, λ - 3 k]

I_{1} = [k, λ - 2 k]

I_{2} = [2 k, λ - k]

I_{3} = [3 k, λ] .

Since vertices

0

2

4

and

6

are two distance apart from each other, the differences between any two of

f (0), f (2), f (4), f (6)

are at least

k

, that is,

f (0), f (2), f (4), f (6) \in ⋃_{i = 0}^{3} I_{i}

. According to the assumption

λ < j + 3 k

, the lengths of interval

I_{i}

is less than

j,

where

i = 0, 1, 2, 3

. For convenience, let

f (0) \in I_{a}

f (2) \in I_{b}

{f (4) \in I}_{c}

{f (6) \in I}_{d}

, where

\{a, b, c, d\} = \{0, 1, 2, 3\}

. Moreover, vertices

2

4

6

and

8

are at distances two from each other, then

f (2), f (4), f (6), f (8) \in ⋃_{i = 0}^{3} I_{i}

. It implies that

f (8), f (0) \in I_{a}

. Furthermore, vertices 1,

3

5

and

7

are also two distances apart mutually, then one of

f (1), f (3), f (5), f (7)

lies in

I_{a}

. However, each of vertices 1,

3

5

and

7

is adjacent to

v_{0}

v_{8}

, then

f (1), f (3), f (5), f (7) \notin I_{a}

which is a contradiction. Thus

λ_{j, k} (G) \geq j + 3 k

Hence,

λ_{j, k} (G) = j + 3 k

Theorem 2.3 Let

n

be an integer and

j, k

be non-negative real numbers with

k \geq 2 j

. For

n \geq 7

σ_{j, k} (G) = 4 k

Proof: Given an labeling

f

for graph

G

as follows.

f (i) = {[\frac{i}{2}]}_{4} k

0 \leq i \leq n - 1

For any vertex

i

, according to the symmetry of the distance graph, it is needed to verify the labels of vertices

i + 1

and

i + 5

which are adjacent to vertex

i

, and the labels of vertices

i + 2, i + 4, i + 6

i + 10

which are 2 distance apart from vertex

i

1. The circular distances between

f (v_{i + 1})

and

f (v_{i})

f (v_{i + 5})

and

f (v_{i})

are shown as follows.

{|f (v_{i + 1}) - f (v_{i})|}_{4 k} = {|{[\frac{i + 1}{2}]}_{4} k - {[\frac{i}{2}]}_{4} k|}_{4 k}

= \{\begin{matrix} {|\frac{k}{2}|}_{4 k} = \min \{\frac{k}{2}, 4 k - \frac{k}{2}\} = \frac{k}{2}, \\ {|\frac{7 k}{2}|}_{4 k} = \min \{\frac{7 k}{2}, 4 k - \frac{7 k}{2}\} = \frac{k}{2}, \end{matrix} \geq j .

{|f (v_{i + 5}) - f (v_{i})|}_{4 k} = {|{[\frac{i + 5}{2}]}_{4} k - {[\frac{i}{2}]}_{4} k|}_{4 k}

= \{\begin{matrix} {|\frac{5 k}{2}|}_{4 k} = \min \{\frac{5 k}{2}, 4 k - \frac{5 k}{2}\} = \frac{3 k}{2}, \\ {|\frac{3 k}{2}|}_{4 k} = \min \{\frac{3 k}{2}, 4 k - \frac{3 k}{2}\} = \frac{3 k}{2}, \end{matrix} \geq j .

2. The circular distances between

f (v_{i + 2})

and

f (v_{i})

f (v_{i + 4})

and

f (v_{i})

f (v_{i + 6})

and

f (v_{i})

f (v_{i + 10})

and

f (v_{i})

are shown as follows.

{|f (v_{i + 2}) - f (v_{i})|}_{4 k} = {|{[\frac{i + 2}{2}]}_{4} k - {[\frac{i}{2}]}_{4} k|}_{4 k}

= \{\begin{matrix} {|k|}_{4 k} = \min \{k, 4 k - k\} = k, \\ {|3 k|}_{4 k} = \min \{3 k, 4 k - 3 k\} = k, \end{matrix} \geq k .

{|f (v_{i + 4}) - f (v_{i})|}_{4 k} = {|{[\frac{i + 4}{2}]}_{4} k - {[\frac{i}{2}]}_{4} k|}_{4 k}

= {|2 k|}_{4 k} = \min \{2 k, 4 k - 2 k\} = 2 k \geq k .

{|f (v_{i + 6}) - f (v_{i})|}_{4 k} = {|{[\frac{i + 6}{2}]}_{4} k - {[\frac{i}{2}]}_{4} k|}_{4 k}

= \{\begin{matrix} {|k|}_{4 k} = \min \{k, 4 k - k\} = k, \\ {|3 k|}_{4 k} = \min \{3 k, 4 k - 3 k\} = k, \end{matrix} \geq k .

{|f (v_{i + 10}) - f (v_{i})|}_{4 k} = {|{[\frac{i + 10}{2}]}_{4} k - {[\frac{i}{2}]}_{4} k|}_{4 k}

= \{\begin{matrix} {|k|}_{4 k} = \min \{k, 4 k - k\} = k, \\ {|3 k|}_{4 k} = \min \{3 k, 4 k - 3 k\} = k, \end{matrix} \geq k .

According to the above verification and Lemma 1.3,

f

is a circular

4 k

L (j, k)

-labeling of distance graph

G

, thus,

σ_{j, k} (G) \leq 4 k

for

k \geq 2 j .

On the other hand, any pair of vertices

0

2

4

and

6

are at distance two, according to the definition of circular

L (j, k)

-labeling, the circular distance between any two labels of vertices

0

2

4

and

6 is at least k,

thus,

σ_{j, k} (G) \geq 4 k

Hence,

σ_{j, k} (G) = 4 k

for

k \geq 2 j .

3. Conclusion

In this article, we mainly investigate the

L (j, k)

-labeling numbers and the circular

L (j, k)

-labeling numbers of distance graph

D_{n} (1, 5)

and obtain the following results.

Let

n

be an integer and

j, k, σ

be non-negative real numbers. Then

λ_{j, k} (G) = \{\begin{matrix} 3 k, & if 7 \leq n \leq 8 . \\ j + 3 k, & if n \geq 9 . \end{matrix}

σ_{j, k} (G) = 4 k

, where

n \geq 7

and

k \geq 2 j

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Shuping, H., Qiong, W. (2025). The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5). International Journal of Systems Science and Applied Mathematics, 10(1), 7-11. https://doi.org/10.11648/j.ijssam.20251001.12

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Shuping, H.; Qiong, W. The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5). Int. J. Syst. Sci. Appl. Math. 2025, 10(1), 7-11. doi: 10.11648/j.ijssam.20251001.12

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Shuping H, Qiong W. The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5). Int J Syst Sci Appl Math. 2025;10(1):7-11. doi: 10.11648/j.ijssam.20251001.12

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@article{10.11648/j.ijssam.20251001.12,
  author = {He Shuping and Wu Qiong},
  title = {The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5)},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {10},
  number = {1},
  pages = {7-11},
  doi = {10.11648/j.ijssam.20251001.12},
  url = {https://doi.org/10.11648/j.ijssam.20251001.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20251001.12},
  abstract = {For j ≤ k, the L(j, k)- labeling problem arose form the code assignment problem of the wireless network. That is, let n,j,k be non-negative real numbers with j ≤ k, an n-L(j, k)-labeling of a graph G is mapping f: V(G)→[0, n] such that |f(u)-f(v)| ≥ j if d(u, v)=1, and |f(u)-f(v)|≥k if t d(u, v)=2. The span of f is the difference between the maximum and minimum labeling numbers assigned by f. The L(j, k)-labeling number of graph G, denoted by λ(j,k) (G), is the minimum span of all L(j, k)-labeling of G. The infinite distance graph, denoted by D (d1,d2,…,dk), has the set Z of integers as a vertex set and in which two vertices i,j∈Z are adjacent if and only if |i-j|∈D. The finite distance graph, denoted by Dn (d1,d2,…,dk), is the subgraph of D(d1,d2,…,dk) induced by vertices {0,1,…,n-1}. This paper determines the L(j, k)-labeling number and the circular L(j, k)-labeling number of distance graph Dn (1,5) for 2j ≤ k.},
 year = {2025}
}

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TY  - JOUR
T1  - The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5)
AU  - He Shuping
AU  - Wu Qiong
Y1  - 2025/02/10
PY  - 2025
N1  - https://doi.org/10.11648/j.ijssam.20251001.12
DO  - 10.11648/j.ijssam.20251001.12
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 7
EP  - 11
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20251001.12
AB  - For j ≤ k, the L(j, k)- labeling problem arose form the code assignment problem of the wireless network. That is, let n,j,k be non-negative real numbers with j ≤ k, an n-L(j, k)-labeling of a graph G is mapping f: V(G)→[0, n] such that |f(u)-f(v)| ≥ j if d(u, v)=1, and |f(u)-f(v)|≥k if t d(u, v)=2. The span of f is the difference between the maximum and minimum labeling numbers assigned by f. The L(j, k)-labeling number of graph G, denoted by λ(j,k) (G), is the minimum span of all L(j, k)-labeling of G. The infinite distance graph, denoted by D (d1,d2,…,dk), has the set Z of integers as a vertex set and in which two vertices i,j∈Z are adjacent if and only if |i-j|∈D. The finite distance graph, denoted by Dn (d1,d2,…,dk), is the subgraph of D(d1,d2,…,dk) induced by vertices {0,1,…,n-1}. This paper determines the L(j, k)-labeling number and the circular L(j, k)-labeling number of distance graph Dn (1,5) for 2j ≤ k.
VL  - 10
IS  - 1
ER  -

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Author Information

He Shuping

School of Science, Tianjin University of Technology and Education, Tianjin, China

Contact Email

http://orcid.org/0009-0004-6297-6524
Wu Qiong

School of Science, Tianjin University of Technology and Education, Tianjin, China

Contact Email

http://orcid.org/0000-0001-6766-2760

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APA Style

Shuping, H., Qiong, W. (2025). The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5). International Journal of Systems Science and Applied Mathematics, 10(1), 7-11. https://doi.org/10.11648/j.ijssam.20251001.12

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ACS Style

Shuping, H.; Qiong, W. The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5). Int. J. Syst. Sci. Appl. Math. 2025, 10(1), 7-11. doi: 10.11648/j.ijssam.20251001.12

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AMA Style

Shuping H, Qiong W. The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5). Int J Syst Sci Appl Math. 2025;10(1):7-11. doi: 10.11648/j.ijssam.20251001.12

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@article{10.11648/j.ijssam.20251001.12,
  author = {He Shuping and Wu Qiong},
  title = {The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5)},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {10},
  number = {1},
  pages = {7-11},
  doi = {10.11648/j.ijssam.20251001.12},
  url = {https://doi.org/10.11648/j.ijssam.20251001.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20251001.12},
  abstract = {For j ≤ k, the L(j, k)- labeling problem arose form the code assignment problem of the wireless network. That is, let n,j,k be non-negative real numbers with j ≤ k, an n-L(j, k)-labeling of a graph G is mapping f: V(G)→[0, n] such that |f(u)-f(v)| ≥ j if d(u, v)=1, and |f(u)-f(v)|≥k if t d(u, v)=2. The span of f is the difference between the maximum and minimum labeling numbers assigned by f. The L(j, k)-labeling number of graph G, denoted by λ(j,k) (G), is the minimum span of all L(j, k)-labeling of G. The infinite distance graph, denoted by D (d1,d2,…,dk), has the set Z of integers as a vertex set and in which two vertices i,j∈Z are adjacent if and only if |i-j|∈D. The finite distance graph, denoted by Dn (d1,d2,…,dk), is the subgraph of D(d1,d2,…,dk) induced by vertices {0,1,…,n-1}. This paper determines the L(j, k)-labeling number and the circular L(j, k)-labeling number of distance graph Dn (1,5) for 2j ≤ k.},
 year = {2025}
}

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TY  - JOUR
T1  - The L(j, k)-Labeling Number and Circular L(j,k)-Labeling Number of Distance Graph Dn(1,5)
AU  - He Shuping
AU  - Wu Qiong
Y1  - 2025/02/10
PY  - 2025
N1  - https://doi.org/10.11648/j.ijssam.20251001.12
DO  - 10.11648/j.ijssam.20251001.12
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 7
EP  - 11
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20251001.12
AB  - For j ≤ k, the L(j, k)- labeling problem arose form the code assignment problem of the wireless network. That is, let n,j,k be non-negative real numbers with j ≤ k, an n-L(j, k)-labeling of a graph G is mapping f: V(G)→[0, n] such that |f(u)-f(v)| ≥ j if d(u, v)=1, and |f(u)-f(v)|≥k if t d(u, v)=2. The span of f is the difference between the maximum and minimum labeling numbers assigned by f. The L(j, k)-labeling number of graph G, denoted by λ(j,k) (G), is the minimum span of all L(j, k)-labeling of G. The infinite distance graph, denoted by D (d1,d2,…,dk), has the set Z of integers as a vertex set and in which two vertices i,j∈Z are adjacent if and only if |i-j|∈D. The finite distance graph, denoted by Dn (d1,d2,…,dk), is the subgraph of D(d1,d2,…,dk) induced by vertices {0,1,…,n-1}. This paper determines the L(j, k)-labeling number and the circular L(j, k)-labeling number of distance graph Dn (1,5) for 2j ≤ k.
VL  - 10
IS  - 1
ER  -

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