EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6601
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Edge Irregular Reflexive Labeling of Ladder Graph
Corona Null Graph Families

Diari Indriati1, Muhammad Yogi Sentosa1, Zhafirah Miranti Verdiana1, Titin
Sri Martini1, Putranto Hadi Utomo1, Nughthoh Arfawi Kurdi1, Isnaini Rosyida2,∗

1 Department of Mathematics, Faculty of Mathematics and Natural Sciences,
Universitas Sebelas Maret, Surakarta, Indonesia
2 Department of Mathematics, Faculty of Mathematics and Natural Sciences,
Universitas Negeri Semarang, Indonesia

Abstract. Given a graph G(V,E) or simply written as G. The graph labeling was first introduced
in 1960, it was a function that mapping integers or labels to graph elements (vertices, edges, or
both of them) which must satisfy some certain criteria. This concepts was applied in some real
problems. In 2007, there was a new concept on labeling, i.e., “vertex irregular total labeling” and
“edge irregular total labeling”. In 2017, there was a new concept, i.e., “vertex irregular reflexive
labeling” and “edge irregular reflexive labeling”. The “edge irregular reflexive k− labeling” of G
is a mapping that puts an even number label from 0 to 2kv to every vertex and a positive integer
label from 1 to ke to every edge with different weights for each edge. The minimum k of the biggest
label among all possible “edge irregular reflexive k− labeling” of G is called the “reflexive edge
strength” of G, denoted as res(G). There are only few result on “vertex irregular reflexive labeling”
or “edge irregular reflexive labeling” of corona product graphs. Therefore, the goal of this study
is to examine the res of corona product of some ladder graphs and null graphs. We get the results
as follows: res(SLn � Nm) for “n ≥ 2 and m ≥ 1” are “

⌈
2nm+3n−3

3

⌉
” for “2nm + 3n − 3 6≡ 2, 3

(mod 6)” and “
⌈
2nm+3n−3

3

⌉
+ 1” for “2nm + 3n − 3 ≡ 2, 3 (mod 6)”. Moreover, res(DLn � Nm)

for n ≥ 2 and m ≥ 1 are “
⌈
2nm+5n−4

3

⌉
” for “2nm+5n− 4 6≡ 2, 3 (mod 6)” and “

⌈
2nm+5n−4

3

⌉
+1”

for “2nm+ 5n− 4 ≡ 2, 3 (mod 6)”. These results contribute to developing the theory of reflexive
labeling.
2020 Mathematics Subject Classifications: 05C78
Key Words and Phrases: Corona, slanting ladder graph, diagonal ladder graph, null graph,
edge irregular reflexive k-labeling

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6601

Email addresses: diari_indri@staff.uns.ac.id (D. Indriati),
isnaini@mail.unnes.ac.id (I. Rosyida)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 2 of 17

1. Introduction

Graph theory is a branch of mathematics that studies relationship between objects
through visual representation in the form of graphs. Graphs are used to represent rela-
tionships in various fields such as biology, social networks, and transportation systems.
According to Diestel [1], A graph consists of a non-empty set of vertices {v1, v2, v3, · · · , vn}
and an edge set {e1, e2, e3, · · · , em}. Every edge has two end vertices to indicate it and is
often depicted as a line segment joining these two vertices.

One of the topics discussed in graph theory is graph labeling. This labeling was first
introduced in 1960 and according to Wallis [2], a mapping called ”graph labeling” gives
each vertex and edge in the graph a label or an integer, which must satisfy some certain
criteria or rules.

The “vertex irregular total k−labeling” and “edge irregular total k− labeling” are two
novel ideas in graph theory that were introduced by Bača et al in 2007 [3]. However, in
2017, this concept was developed again by Ryan et al [4] by introducing “vertex irregular
reflexive k-labeling” and “edge irregular reflexive k labeling”. In G, a reflexive edge irreg-
ular k-labeling is defined as a function that can put an even number label from 0 to 2kv to
each vertex and a positive integer label from 1 to ke to each edge where k=max{ke, 2kv}
with a different weight for each edge. The reflexive edge strength, represented by res(G),
is the lowest value k of the largest label [4]. The following lower bound of res(G) was
given by Ryan et al [4].
Lemma 1. Considering each graph G,

res(G) ≥


⌈
|E(G)|

3

⌉
, if |E(G)| 6≡ 2, 3 (mod 6),⌈

|E(G)|
3

⌉
+ 1, if |E(G)| ≡ 2, 3 (mod 6).

The “edge irregular reflexive k-labeling” of several graphs have been investigated, in-
cludes Tanna et al. [5] proved of prism graph (Dn), wheel graph (Wn), fan graph (Fn)
and basket graph (Bn). Budi et al. [6] investigated cycles graphs (Cn). Bača et al. [3]
examined tadpole graphs Tm,1 and Tm,2. Indriati et al. found res of “corona of path” and
other graphs [7]. Agustin et al. provided the res of some trees and some almost regular
graphs [8, 9]. Santoso et al. constructed an algorithm for the non-inclusive vertex irregular
labeling [10]. For more results on the res of various graphs, the readers could see [11].

In this research, we focuss on the families related to ladder graphs, i.e., slanting ladder
and diagonal ladder [12],[13]. The reflexive edge strength of slanting ladder corona null
graph (SLn�Nm), n ≥ 2 and m ≥ 1 and diagonal ladder corona null graph (DLn�Nm),
n ≥ 2 and m ≥ 1 will be determined.

2. Main Results

2.1. Corona of Slanting Ladder and Null Graph

The corona of “slanting ladder” and “null graph”, symbolized by SLn � Nm, is a
connected graph with V (SLn �Nm) = {ui, vi : 1 ≤ i ≤ n} ∪ {ui,j , vi,j : 1 ≤ i ≤ n, 1 ≤ j ≤



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 3 of 17

m} and E(SLn �Nm) = {uiui+1, vivi+1, uivi+1 : 1 ≤ i ≤ n − 1} ∪ {vivi,j , uiui,j : 1 ≤ i ≤
n, 1 ≤ j ≤ m}. Therefore, this graph has order 2nm + 2n and size 2nm + 3n − 3. This
graph is shown in Figure 1.

Figure 1: The corona graph SLn �Nm

The res of SLn �Nm is proved in Theorem 1.

Theorem 1. Given SLn �Nm for n ≥ 2 and m ≥ 1,

res(SLn �Nm) =

{⌈
2nm+3n−3

3

⌉
, if 2nm+ 3n− 3 6≡ 2, 3 (mod 6),⌈

2nm+3n−3
3

⌉
+ 1, if 2nm+ 3n− 3 ≡ 2, 3 (mod 6).

(1)

Proof. Since the size is 2nm+ 3n− 3, then by using Lemma 1 we get:

res(SLn �Nm) ≥

{⌈
2nm+3n−3

3

⌉
, if 2nm+ 3n− 3 6≡ 2, 3 (mod 6),⌈

2nm+3n−3
3

⌉
+ 1, if 2nm+ 3n− 3 ≡ 2, 3 (mod 6).

(2)

Statement (2) is a lower bound of res(SLn � Nm). Further, we verify the upper
bound of the res. Construct θ as k-labeling of corona of slanting ladder and null graph
with k =

⌈
2nm+3n−3

3

⌉
for 2nm + 3n − 3 6≡ 2, 3 (mod 6) and k =

⌈
2nm+3n−3

3

⌉
+ 1 for

2nm+ 3n− 3 ≡ 2, 3 (mod 6) below.



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 4 of 17

Label of vertex vi for 1 ≤ i ≤ n,

θ(vi) =



0, i = 1,
2m+6−2a

3 , i = 2, m ≡ a (mod 3), a = 0, 1, 2,
8m+6

3 , i = 4, m ≡ 0 (mod 3),
10m+10

3 , i = 5, m ≡ 2 (mod 3),
2im+3i

3 , i ≡ 0 (mod 6),

i ≡ 2 (mod 6), m ≡ 0 (mod 3), i 6= 2,
i ≡ 4 (mod 3), m ≡ 0 (mod 3), i 6= 4,

2im+3i−3
3 , i ≡ 1 (mod 6), m ≡ 0 (mod 3),

i ≡ 3 (mod 6),
i ≡ 5 (mod 3), m ≡ 0 (mod 3),

2im+3i−4
3 , i ≡ 2 (mod 6), m ≡ 1 (mod 3), i 6= 2

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
2im+3i−1

3 , i ≡ 1 (mod 6), m ≡ 2 (mod 3), i 6= 1

i ≡ 5 (mod 6), m ≡ 1 (mod 3),
2im+3i−2

3 , i ≡ 2 (mod 6), m ≡ 2 (mod 3), i 6= 2

i ≡ 4 (mod 6), m ≡ 1 (mod 3),
2im+3i+1

3 , i ≡ 1 (mod 6), m ≡ 1 (mod 3), i 6= 1,
i ≡ 5 (mod 6), m ≡ 2 (mod 3), i 6= 5.

The label of ui for i = 1, 2, · · · , n:

θ(ui) =



0, i = 1,
4m
3 , i = 2, m ≡ 0 (mod 3),
8m+6

3 , i = 4, m ≡ 0 (mod 3),
10m+10

3 , i = 5, m ≡ 2 (mod 3),
2im+3i

3 , i ≡ 0 (mod 6),
i ≡ 2 (mod 6), m ≡ 0 (mod 3), i 6= 2,
i ≡ 4 (mod 6), m ≡ 0 (mod 3),‘i 6= 4,

2im+3i−3
3 , i ≡ 1 (mod 6), m ≡ 0 (mod 3), i 6= 1,

i ≡ 3 (mod 6),
i ≡ 5 (mod 6), m ≡ 0 (mod 3),

2im+3i−4
3 , i ≡ 2 (mod 6), m ≡ 1 (mod 3),

i ≡ 4 (mod 6), m ≡ 2 (mod 3),



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 5 of 17

Continue of label of ui for i = 1, 2, · · · , n:

θ(ui) =



2im+3i−1
3 , i ≡ 1 (mod 6), m ≡ 2 (mod 3), i 6= 1

i ≡ 2 (mod 6), m ≡ 1 (mod 3),
2im+3i−2

3 , i ≡ 2 (mod 6), m ≡ 2 (mod 3),
i ≡ 4 (mod 3), m ≡ 1 (mod 3),

2im+3i+1
3 , i ≡ 1 (mod 6), m ≡ 1 (mod 3), i 6= 1

i ≡ 5 (mod 6), m ≡ 2 (mod 3), i 6= 5.

Label of vi,j for 1 ≤ i ≤ n and 1 ≤ j ≤ m:

θ(vi,j) =



0, i = 1,
4m
3 , i = 2, m ≡ 0 (mod 3),
2im+3i

3 , i ≡ 0 (mod 6),
i ≡ 2 (mod 3), m ≡ 0 (mod 3),
i ≡ 4 (mod 6), m ≡ 0 (mod 3), i 6= 2,

2im+3i−3
3 , i ≡ 1 (mod 6), m ≡ 0 (mod 3),

i ≡ 3 (mod 6),
i ≡ 5 (mod 6), m ≡ 0 (mod 3),

2im+3i−4
3 , i ≡ 2 (mod 6), m ≡ 1 (mod 3),

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
2im+3i−1

3 , i ≡ 1 (mod 6), m ≡ 2 (mod 3), i 6= 1

i ≡ 5 (mod 6), m ≡ 1 (mod 3),
2im+3i−2

3 , i ≡ 2 (mod 6), m ≡ 2 (mod 3),
i ≡ 4 (mod 6), m ≡ 1 (mod 3),

2im+3i+1
3 , i ≡ 1 (mod 6), m ≡ 1 (mod 3), i 6= 1,

i ≡ 5 (mod 6), m ≡ 2 (mod 3).

Label of ui,j for 1 ≤ i ≤ n and j = 1, 2, · · · ,m:

θ(ui,j) =



2m−2
3 , i = 1, m ≡ 1 (mod 3),

2m−4
3 , i = 1, m ≡ 2 (mod 3),

2im+3i
3 , i ≡ 0 (mod 6),

i ≡ 2 (mod 6), m ≡ 0 (mod 3),
i ≡ 4 (mod 6), m ≡ 0 (mod 3),



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 6 of 17

θ(ui,j) =



2im+3i−3
3 , i ≡ 1 (mod 6), m ≡ 0 (mod 3),

i ≡ 3 (mod 6),
i ≡ 5 (mod 6), m ≡ 0 (mod 3),

2im+3i−4
3 , i ≡ 2 (mod 6), m ≡ 1 (mod 3),

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
2im+3i−1

3 , i ≡ 1 (mod 6), m ≡ 2 (mod 3), i 6= 1

i ≡ 5 (mod 6), m ≡ 1 (mod 3),
2im+3i−2

3 , i ≡ 2 (mod 6), m ≡ 2 (mod 3),
i ≡ 4 (mod 6), m ≡ 1 (mod 3),

2im+3i+1
3 , i ≡ 1 (mod 6), m ≡ 1 (mod 3), i 6= 1,

i ≡ 5 (mod 6), m ≡ 2 (mod 3).

Label of edge vivi+1 for i = 1, 2, · · · , n− 1,

θ(vivi+1) =



4m+2a−3
3 , i = 1, m ≡ a (mod 3), a = 0, 1, 2,

4m+2a
3 , i = 2, m ≡ a (mod 3), a = 0, 1, 2,

4m+9
3 , i = 3, m ≡ 0 (mod 3),

8m+11
3 , i = 5, m ≡ 2 (mod 3),

(2i−2)m+3i−6
3 , i ≡ 0, 3 (mod 6), m ≡ 0 (mod 3), i 6= 3,

i ≡ 1, 4 (mod 6), i 6= 1,
i ≡ 2, 5 (mod 6), m ≡ 0 (mod 3), i 6= 2,

(2i−2)m+3i−8
3 , i ≡ 0 (mod 6), m ≡ 2 (mod 3),

i ≡ 5 (mod 6), m ≡ 1 (mod 3),
(2i−2)m+3i−2

3 , i ≡ 2 (mod 6), m ≡ 1 (mod 3), i 6= 2

i ≡ 3 (mod 6), m ≡ 2 (mod 3),
(2i−2)m+3i−4

3 , i ≡ 2 (mod 6), m ≡ 2 (mod 3), i 6= 2

i ≡ 3 (mod 6), m ≡ 1 (mod 3),
(2i−2)m+3i−10

3 , i ≡ 0 (mod 6), m ≡ 1 (mod 3),
i ≡ 5 (mod 6), m ≡ 2 (mod 3), i 6= 5.

Label of edge uiui+1 where i = 1, 2, · · · , n− 1,

θ(uiui+1) =


2m+9−2a

3 , i = 1, m ≡ a (mod 3), a = 0, 1, 2,
2m+12

3 , i = 2, m ≡ 0 (mod 3),
4m+15

3 , i = 3, m ≡ 0 (mod 3),
6m+18

3 , i = 4, m ≡ 0, 2 (mod 3),



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 7 of 17

Continue to label of uiui+1:

θ(uiui+1) =



2m+9−2a
3 , i = 1, m ≡ a (mod 3), a = 0, 1, 2,

2m+12
3 , i = 2, m ≡ 0 (mod 3),

4m+15
3 , i = 3, m ≡ 0 (mod 3),

6m+18
3 , i = 4, m ≡ 0, 2 (mod 3),

8m+17
3 , i = 5, m ≡ 2 (mod 3),

(2i−2)m+3i
3 , i ≡ 0, 3 (mod 6), i 6= 3,

i ≡ 1, 4 (mod 6), i 6= 1, 4

i ≡ 2, 5 (mod 6), m ≡ 0 (mod 3), i 6= 2,
(2i−2)m+3i−2

3 , i ≡ 0 (mod 6), m ≡ 2 (mod 3),
i ≡ 5 (mod 6), m ≡ 1 (mod 3),

(2i−2)m+3i+4
3 , i ≡ 2 (mod 6), m ≡ 1 (mod 3),

i ≡ 3 (mod 6), m ≡ 2 (mod 3),
(2i−2)m+3i+2

3 , i ≡ 2 (mod 6), m ≡ 2 (mod 3),
i ≡ 3 (mod 6), m ≡ 1 (mod 3),

(2i−2)m+3i−4
3 , i ≡ 0 (mod 6), m ≡ 1 (mod 3),

i ≡ 5 (mod 6), m ≡ 2 (mod 3), i 6= 5.

Label of edge uivi+1 for i = 1, 2, · · · , n− 1:

θ(uivi+1) =



4m+2a
3 , i = 1, m ≡ a (mod 3), a = 0, 1, 2,

2m+9
3 , i = 2, m ≡ 0 (mod 3),

4m+12
3 , i = 3, m ≡ 0 (mod 3),

8m+14
3 , i = 5, m ≡ 2 (mod 3),

(2i−2)m+3i−3
3 , i ≡ 0, 3 (mod 6), m ≡ 0 (mod 3), i 6= 3,

i ≡ 1, 4 (mod 6), i 6= 1

i ≡ 2, 5 (mod 6), m ≡ 0 (mod 3), i 6= 2,
(2i−2)m+3i+1

3 , i ≡ 2 (mod 6), m ≡ 1 (mod 3),
i ≡ 3 (mod 6), m ≡ 2 (mod 3),

(2i−2)m+3i−5
3 , i ≡ 0 (mod 6), m ≡ 2 (mod 3),

i ≡ 5 (mod 6), m ≡ 1 (mod 3),
(2i−2)m+3i−1

3 , i ≡ 2 (mod 6), m ≡ 2 (mod 3),
i ≡ 3 (mod 6), m ≡ 1 (mod 3),

(2i−2)m+3i−7
3 , i ≡ 0 (mod 6), m ≡ 1 (mod 3),

i ≡ 5 (mod 6), m ≡ 2 (mod 3), i 6= 5.



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 8 of 17

Label of edge vivi,j for i = 1, 2, · · · , n and j = 1, 2, · · · ,m:

θ(vivi,j) =



i+ j − 1, i = 1, 2, 3,
(2m+12

3 ) + j − 1, i = 4, m ≡ 0 (mod 3),
(4m+13

3 ) + j − 1, i = 5, m ≡ 2 (mod 3),
( (2i−6)m+3i

3 ) + j − 1, i ≡ 1 (mod 6), m ≡ 0 (mod 3), i 6= 1,
i ≡ 0 (mod 6),
i ≡ 5 (mod 6), m ≡ 0 (mod 3),

( (2i−6)m+3i−6
3 ) + j − 1, i ≡ 0 (mod 6),

i ≡ 2 (mod 6), m ≡ 0 (mod 3), i 6= 2,
i ≡ 4 (mod 6), m ≡ 0 (mod 3), i 6= 4,

( (2i−6)m+3i−2
3 ) + j − 1, i ≡ 2 (mod 6), m ≡ 2 (mod 3), i 6= 2,

i ≡ 4 (mod 6), m ≡ 1 (mod 3),
( (2i−6)m+3i−8

3 ) + j − 1, i ≡ 1 (mod 6), m ≡ 1 (mod 3), i 6= 1

i ≡ 5 (mod 6), m ≡ 2 (mod 3), i 6= 5,
( (2i−6)m+3i+2

3 ) + j − 1, i ≡ 2 (mod 6), m ≡ 1 (mod 3), i 6= 2,
i ≡ 4 (mod 6), m ≡ 2 (mod 3),

( (2i−6)m+3i−4
3 ) + j − 1, i ≡ 1 (mod 6), m ≡ 2 (mod 3), i 6= 1

i ≡ 5 (mod 6), m ≡ 1 (mod 3).

Label of uiui,j where i = 1, 2, · · · , n and j = 1, 2, · · · ,m:

θ(uiui,j) =



(m+2a+3
3 ) + j − 1, i = 1, m ≡ a (mod 3), a = 1, 2, 3,

(m+6
3 ) + j − 1, i = 2, m ≡ 0 (mod 3),

(5m+12
3 ) + j − 1, i = 4, m ≡ 0 (mod 3),

(7m+13
3 ) + j − 1, i = 5, m ≡ 2 (mod 3),

( (2i−3)m+3i−2
3 ) + j − 1, i ≡ 4 (mod 6), m ≡ 1 (mod 3),

i ≡ 5 (mod 6), m ≡ 2 (mod 3),
( (2i−3)m+3i−8

3 ) + j − 1, i ≡ 1 (mod 3), m ≡ 1 (mod 3),
i ≡ 2 (mod 6), m ≡ 2 (mod 3), i 6= 2,

( (2i−3)m+3i+2
3 ) + j − 1, i ≡ 2 (mod 6), m ≡ 1 (mod 3),

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
( (2i−3)m+3i−4

3 ) + j − 1, i ≡ 1 (mod 6), m ≡ 2 (mod 3),
i ≡ 5 (mod 6), m ≡ 1 (mod 3), i 6= 2,

( (2i−3)m+3i−6
3 ) + j − 1, i ≡ 0 (mod 6), m ≥ 1,

i ≡ 2 (mod 6), m ≡ 0 (mod 3), i 6= 2,
i ≡ 4 (mod 6), m ≡ 0 (mod 3), i 6= 4,



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 9 of 17

Continue of label of edge uiui,j where 1 ≤ i ≤ n and 1 ≤ j ≤ m:

θ(uiui,j) =

(
(2i− 3)m+ 3i

3

)
+ j − 1,

for i ≡ 3 (mod 6),m ≥ 1; i ≡ 1 (mod 6),m ≡ 0 (mod 3); and i ≡ 2 (mod 6),m ≡ 0
(mod 3).

Based on the above labeling, the maximum value of the labels is
⌈
2nm+3n−3

3

⌉
when

2nm + 3n − 3 6≡ 2, 3 (mod 6) and
⌈
2nm+3n−3

3

⌉
+ 1 when 2nm + 3n − 3 ≡ 2, 3 (mod 6).

The weight of an edge uv is wtθ(uv) = θ(u) + θ(v) + θ(uv). We get the edge weights as
follows:

wtθ(vivi+1) = 2im+ 3i− 2, 1 ≤ i ≤ n− 1.
wtθ(uiui+1) = 2im+ 3i, 1 ≤ i ≤ n− 1.
wtθ(uivi+1) = 2im+ 3i− 1, 1 ≤ i ≤ n− 1.
wtθ(v1v1,j) = j,

wtθ(vivi,j) = ((2i− 2)m+ 3i− 2) + j − 1, 2 ≤ i ≤ n, 1 ≤ j ≤ m.
wtθ(uiui,j) = ((2i− 2)m+ 3i− 2) + j − 1, 1 ≤ i ≤ n, 1 ≤ j ≤ m.

The weights of all edges on the corona SLn�Nm are distinct. We get the lower bound
of res(SLn�Nm) same as this upper bound. Then, res(SLn�Nm) is obtained. Therefore
θ satisfies the edge irregular reflexive k-labeling and the res is shown in Theorem 1. It
completes the proof.

Figure 2 gives the illustration of SL2 �N2-labeling.

Figure 2: The edge irregular reflexive 4-labeling of SL2 �N2

On the labeling, the label of each edge and each vertex is shown with blue numbers. A
number in red indicates the weight of each edge.



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 10 of 17

2.2. Corona of Diagonal Ladder and Null Graph

Corona of “diagonal ladder” and “null graph”, symbolized by DLn � Nm, is a
connected graph with V (DLn �Nm) = {ui, vi : 1 ≤ i ≤ n} ∪ {ui,j , vi,j : 1 ≤ i ≤ n, 1 ≤ j ≤
m} and E(DLn�Nm) = {uiui+1, vivi+1, uivi+1, viui+1 : 1 ≤ i ≤ n−1}∪{uivi : 1 ≤ i ≤ n}∪
{uiui,j , vivi,j : 1 ≤ i ≤ n, 1 ≤ j ≤ m}. The order of this graph is = 2nm+2n and the size
is 2nm+ 5n− 4. Figure 3 gives an illustration of DLn �Nm.

Figure 3: Corona of “diagonal ladder” and “null graph” DLn �Nm

The reflexive edge strength of DLn �Nm is presented in Theorem 2.

Theorem 2. For DLn �Nm with n ≥ 2 and m ≥ 1,

res(DLn �Nm) =

{⌈
2nm+5n−4

3

⌉
, for 2nm+ 5n− 4 6≡ 2, 3 (mod 6),⌈

2nm+5n−4
3

⌉
+ 1, for 2nm+ 5n− 4 ≡ 2, 3 (mod 6).

(3)

Proof. Since |E| of DLn � Nm is 2nm + 5n − 4, then using Lemma 1, obtained the
lower bound of res(DLn �Nm)

res(DLn �Nm) ≥

{⌈
2nm+5n−4

3

⌉
, 2nm+ 5n− 4 6≡ 2, 3 (mod 6),⌈

2nm+5n−4
3

⌉
+ 1, 2nm+ 5n− 4 ≡ 2, 3 (mod 6).

(4)

We will prove the upper bound of res(DLn � Nm) for n ≥ 2 and m ≥ 1 by con-
structing a k-label θ of corona of diagonal ladder and null graph with k =

⌈
2nm+5n−4

3

⌉
for

2nm+ 5n− 4 6≡ 2, 3 (mod 6) and k =
⌈
2nm+5n−4

3

⌉
+ 1 for 2nm+ 5n− 4 ≡ 2, 3 (mod 6).



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 11 of 17

Vertex label ui for i = 1, 2, · · · , n:

θ(ui) =



0, i = 1,
4m+2a

3 , i = 2, m ≡ a (mod 3), a = 0, 1, 2,
2im+5i

3 , i ≡ 0 (mod 6), m ≥ 1

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
2im+5i−2a+1

3 , i ≡ 1 (mod 6), i 6= 1, m ≡ a (mod 3), a = 1, 2, 3,
2im+5i+2a−4

3 , i ≡ 2 (mod 6), i 6= 2, m ≡ a (mod 3), a = 0, 1, 2,
2im+5i−3

3 , i ≡ 3 (mod 6), m ≥ 1,
2im+5i−2

3 , i ≡ 4 (mod 6), m ≡ 0 (mod 3),
2im+5i−4

3 , i ≡ 4 (mod 6), m ≡ 1 (mod 3),
2im+5i+2a−7

3 , i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3.

Vertex label vi with 1 ≤ i ≤ n:

θ(vi) =



0, i = 1,
2im+5i

3 , i ≡ 0 (mod 6), m ≥ 1

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
2im+5i−2a+1

3 , i ≡ 1 (mod 6), i 6= 1, m ≡ a (mod 3), a = 1, 2, 3,
2im+5i+2a−4

3 , i ≡ 2 (mod 6), m ≡ a (mod 3), a = 0, 1, 2,
2im+5i−3

3 , i ≡ 3 (mod 6), m ≥ 1,
2im+5i−2

3 , i ≡ 4 (mod 6), m ≡ 0 (mod 3),
2im+5i−4

3 , i ≡ 4 (mod 6), m ≡ 1 (mod 3),
2im+5i+2a−7

3 , i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3.

Vertex label ui,j for i = 1, 2, · · · , n and j = 1, 2, · · · ,m:

θ(ui,j) =



0, i = 1,
2m+6

3 , i = 2, m ≡ 0 (mod 3),
2m+4

3 , i = 2, m ≡ 1 (mod 3),
2m+8

3 , i = 2, m ≡ 2 (mod 3),
2im+5i

3 , i ≡ 0 (mod 6), m ≥ 1

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
2im+5i−2a+1

3 , i ≡ 1 (mod 6), i 6= 1, m ≡ a (mod 3), a = 1, 2, 3,
2im+5i+2a−4

3 , i ≡ 2 (mod 6), i 6= 2, m ≡ a (mod 3), a = 0, 1, 2,
2im+5i−3

3 , i ≡ 3 (mod 6), m ≥ 1,
2im+5i−2

3 , i ≡ 4 (mod 6), m ≡ 0 (mod 3),
2im+5i−4

3 , i ≡ 4 (mod 6), m ≡ 1 (mod 3),
2im+5i+2a−7

3 , i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3.



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 12 of 17

Vertex label vi,j for i = 1, 2, · · · , n and j = 1, 2, · · · ,m:

θ(vi,j) =



2m−2
3 , i = 1, m ≡ 1 (mod 3),

4, i = 2, m = 1,
4m+2

3 , i = 2, m ≡ 1 (mod 3), m 6= 1
2im+5i

3 , i ≡ 0 (mod 6), m ≥ 1

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
2im+5i−2a+1

3 , i ≡ 1 (mod 6), m ≡ a (mod 3), a = 1, 2, 3,
2im+5i+2a−4

3 , i ≡ 2 (mod 6), m ≡ a (mod 3), a = 0, 2, 1 and i 6= 2,
2im+5i−3

3 , i ≡ 3 (mod 6), m ≥ 1,
2im+5i−2

3 , i ≡ 4 (mod 6), m ≡ 0 (mod 3),
2im+5i−4

3 , i ≡ 4 (mod 6), m ≡ 1 (mod 3),
2im+5i+2a−7

3 , i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3.

Edge label uiui+1 for i = 1, 2, · · · , n− 1:

θ(uiui+1) =



2m−2a+6
3 , i = 1, m ≡ a (mod 3), a = 0, 1, 2,

2m−2a+9
3 , i = 2, m ≡ a (mod 3), a = 0, 1, 2,

(2i−2)m+5i+2a−15
3 , i ≡ 0 (mod 6), m ≡ a (mod 3), a = 1, 2, 3,

(2i−2)m+5i−5
3 , i ≡ 1 (mod 6), i 6= 1, m ≡ 0 (mod 3)

i ≡ 4 (mod 6), m ≡ 1 (mod 3),
(2i−2)m+5i−11

3 , i ≡ 1 (mod 6), m ≡ 1, 2 (mod 3), i 6= 1,
i ≡ 3 (mod 6), m ≡ 2 (mod 3),
i ≡ 4 (mod 6), m ≡ 0, 2 (mod 3),

(2i−2)m+5i−2a−7
3 , i ≡ 2 (mod 6), i 6= 2, m ≡ a (mod 3), a = 0, 1, 2

i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3,
(2i−2)m+5i−9

3 , i ≡ 3 (mod 6), m ≡ 0 (mod 3),
(2i−2)m+5i−7

3 , i ≡ 3 (mod 6), m ≡ 1 (mod 3).



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 13 of 17

Edge label vivi+1 for i = 1, 2, · · · , n− 1:

θ(vivi+1) =



2m−2a+9
3 , i = 1, m ≡ a (mod 3), a = 0, 1, 2,

(2i−2)m+5i+2a−6
3 , i ≡ 0 (mod 6), m ≡ a (mod 3), a = 1, 2, 3,

(2i−2)m+5i+4
3 , i ≡ 1 (mod 6), i 6= 1, m ≡ 0 (mod 3)

i ≡ 4 (mod 6), m ≡ 1 (mod 3),
(2i−2)m+5i−2

3 , i ≡ 1 (mod 6), i 6= 1, m ≡ 1, 2 (mod 3),
i ≡ 3 (mod 6), m ≡ 2 (mod 3),
i ≡ 4 (mod 6), m ≡ 0, 2 (mod 3),

(2i−2)m+5i−2a+2
3 , i ≡ 2 (mod 6), m ≡ a (mod 3), a = 0, 1, 2,

i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3,
(2i−2)m+5i

3 , i ≡ 3 (mod 6), m ≡ 0 (mod 3),
(2i−2)m+5i+2

3 , i ≡ 3 (mod 6), m ≡ 1 (mod 3).

Edge label uivi for i = 1, 2, · · · , n:

θ(uivi) =



m+ 1, i = 1,
m−4a+12

3 , i = 2, m ≡ a (mod 3), a = 0, 1, 2,
(2i−3)m+5i−12

3 , i ≡ 0 (mod 6), m ≥ 1

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
(2i−3)m+5i+4a−14

3 , i ≡ 1 (mod 6), i 6= 1, m ≡ a (mod 3), a = 1, 2, 3,
(2i−3)m+5i−4a−4

3 , i ≡ 2 (mod 6), i 6= 2, m ≡ a (mod 3), a = 0, 1, 2,
(2i−3)m+5i−6

3 , i ≡ 3 (mod 6), m ≥ 1,
(2i−3)m+5i−8

3 , i ≡ 4 (mod 6), m ≡ 0 (mod 3),
(2i−3)m+5i−4

3 , i ≡ 4 (mod 6), m ≡ 1 (mod 3),
(2i−3)m+5i−4a+2

3 , i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3.

Edge label uivi+1 for i = 1, 2, · · · , n− 1:

θ(uivi+1) =



2m−2a+3
3 , i = 1, m ≡ a (mod 3), a = 0, 1, 2,

2m−2a+12
3 , i = 2, m ≡ a (mod 3), a = 0, 1, 2,

(2i−2)m+5i+2a−12
3 , i ≡ 0 (mod 6), m ≡ a (mod 3), a = 1, 2, 3,

(2i−2)m+5i−2
3 , i ≡ 1 (mod 6), i 6= 1, m ≡ 0 (mod 3)

i ≡ 4 (mod 6), m ≡ 1 (mod 3),
(2i−2)m+5i−8

3 , i ≡ 1 (mod 6), i 6= 1, m ≡ 1, 2 (mod 3),
i ≡ 3 (mod 6), m ≡ 2 (mod 3),
i ≡ 4 (mod 6), m ≡ 0, 2 (mod 3),



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 14 of 17

Continue of edge label uivi+1:

θ(uivi+1) =


(2i−2)m+5i−2a−4

3 , i ≡ 2 (mod 6), i 6= 2, m ≡ a (mod 3), a = 0, 1, 2,

i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3,
(2i−2)m+5i−6

3 , i ≡ 3 (mod 6), m ≡ 0 (mod 3),
(2i−2)m+5i−4

3 , i ≡ 3 (mod 6), m ≡ 1 (mod 3).

Edge label viui+1 for i = 1, 2, · · · , n− 1:

θ(viui+1) =



2m−2a+12
3 , i = 1, m ≡ a (mod 3), a = 0, 1, 2,

(2i−2)m+5i+2a−9
3 , i ≡ 0 (mod 6), m ≡ a (mod 3), a = 1, 2, 3,

(2i−2)m+5i+1
3 , i ≡ 1 (mod 6), i 6= 1, m ≡ 0 (mod 3)

i ≡ 4 (mod 6), m ≡ 1 (mod 3),
(2i−2)m+5i−5

3 , i ≡ 1 (mod 6), m ≡ 1, 2 (mod 3), i 6= 1,
i ≡ 3 (mod 6), m ≡ 2 (mod 3),
i ≡ 4 (mod 6), m ≡ 0, 2 (mod 3),

(2i−2)m+5i−2a−1
3 , i ≡ 2 (mod 6), m ≡ a (mod 3), a = 0, 1, 2

i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3,
(2i−2)m+5i−3

3 , i ≡ 3 (mod 6), m ≡ 0 (mod 3),
(2i−2)m+5i−1

3 , i ≡ 3 (mod 6), m ≡ 1 (mod 3).

Edge label uiui,j for i = 1, 2, · · · , n and j = 1, 2, · · · ,m:

θ(uiui,j) =



j, i = 1,
j + 3, i = 2, m ≡ 0, 1 (mod 3),
j + 1, i = 2, m ≡ 2 (mod 3),
j + 2, i = 3,
( (2i−6)m+5i−15

3 ) + j, i ≡ 0 (mod 6), m ≥ 1,
i ≡ 4 (mod 6), m ≡ 2 (mod 3),

( (2i−6)m+5i+4a−17
3 ) + j, i ≡ 1 (mod 6), m ≡ a (mod 3), a = 1, 2, 3, i 6= 1,

( (2i−6)m+5i−4a−7
3 ) + j, i ≡ 2 (mod 6), m ≡ a (mod 3), a = 0, 1, 2, i 6= 2,

( (2i−6)m+5i−9
3 ) + j, i ≡ 3 (mod 6), i 6= 3, m ≥ 1.

( (2i−6)m+5i−11
3 ) + j, i ≡ 4 (mod 6), m ≡ 0 (mod 3),

( (2i−6)m+5i−7
3 ) + j, i ≡ 4 (mod 6), m ≡ 1 (mod 3),

( (2i−6)m+5i−4a−1
3 ) + j, i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3.



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 15 of 17

Edge label vivi,j with i = 1, 2, · · · , n and j = 1, 2, · · · ,m:

θ(vivi,j) =



(m+3
3 ) + j, i = 1, m ≡ 0 (mod 3),

(m+5
3 ) + j, i = 1, m ≡ 1 (mod 3),

(m+1
3 ) + j, i = 1, m ≡ 2 (mod 3),

2, i = 2, m = 1,
(m+8

3 ) + j, i = 2, m ≡ 1 (mod 3), m 6= 1

( (2i−3)m+5i−12
3 ) + j, i ≡ 0 (mod 6), m ≥ 1

i ≡ 4 (mod 6), m ≡ 2 (mod 3),
( (2i−3)m+5i+4a−14

3 ) + j, i ≡ 1 (mod 6), m ≡ a (mod 3), a = 1, 2, 3, i 6= 1,
( (2i−3)m+5i−4a−4

3 ) + j, i ≡ 2 (mod 6), m ≡ a (mod 3), a = 0, 1, 2,, i 6= 2,
( (2i−3)m+5i−6

3 ) + j, i ≡ 3 (mod 6), m ≥ 1,
( (2i−3)m+5i−8

3 ) + j, i ≡ 4 (mod 6), m ≡ 0 (mod 3),
( (2i−3)m+5i−4

3 ) + j, i ≡ 4 (mod 6), m ≡ 1 (mod 3),
( (2i−3)m+5i−4a+2

3 ) + j, i ≡ 5 (mod 6), m ≡ a (mod 3), a = 1, 2, 3.

The weight of an edge uv is wtθ(uv) = θ(u) + θ(v) + θ(uv). According to the above
labeling, the edge weights are as follows:

wtθ(uiui+1) = 2im+ 5i− 3, for 1 ≤ i ≤ n− 1.
wtθ(vivi+1) = 2im+ 5i, for 1 ≤ i ≤ n− 1.
wtθ(uivi) = (2i− 1)m+ 5i− 4, for 2 ≤ i ≤ n.

wtθ(uivi+1) = 2im+ 5i− 2, for 1 ≤ i ≤ n− 1.
wtθ(viui+1) = 2im+ 5i− 1, for 1 ≤ i ≤ n− 1.

wtθ(uiui,j) =

{
j, for i = 1, 1 ≤ j ≤ m,
(2i− 2)m+ 5i− 5 + j, for 2 ≤ i ≤ n, 1 ≤ j ≤ m.

wtθ(vivi,j) = (2i− 1)m+ 5i− 4 + j, for 1 ≤ i ≤ n, 1 ≤ j ≤ m.

It could be seen that all edge weights are different, the lower bound and upper bound are
the same as res(DLn �Nm). Therefore θ is the “edge irregular reflexive k-labeling” and
the formula of the res according to Theorem 2. Thus, the theorem is proven.

Figure 4 gives an example of this labeling. The label of each edge is indicated by blue
colour and each vertex label is indicated by green colour. The weight of each edge is
indicated by red colour.



D. Indriati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6601 16 of 17

Figure 4: The edge irregular reflexive 6-labeling of DL2 �N2

3. Conclusion

We have proved the “reflexive edge strength” of ladder graph corona null graph
families SLn �Nm and DLn �Nm as follows:

res(SLn �Nm) =

{⌈
2nm+3n−3

3

⌉
, if 2nm+ 3n− 3 6≡ 2, 3 (mod 6),⌈

2nm+3n−3
3

⌉
+ 1, if 2nm+ 3n− 3 ≡ 2, 3 (mod 6).

Moreover,

res(DLn �Nm) =

{⌈
2nm+5n−4

3

⌉
, 2nm+ 5n− 4 6≡ 2, 3 (mod 6),⌈

2nm+5n−4
3

⌉
+ 1, 2nm+ 5n− 4 ≡ 2, 3 (mod 6).

For the future research, we propose the open problem as follows: how is the reflexive
edge strength for other graph corona null graph.

Acknowledgements

This work was supported by the Fundamental Research Project of DRTPM DIKTI,
LPPM Universitas Sebelas Maret under grant number 1076.1/UN27.22/PT.01.03/2024.
The authors highly appreciate to the editor and reviewers for their suggestions to make
the paper better.

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