EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5673
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Total Edge Irregularity Strength of Star Snake Graphs

Hala Attiya1,∗, Nasr Ahmed2,3, Fatma Salama4

1 Basic Science Department, Faculty of Technology and Education, Beni-Suef University,
Egypt
2 Mathematics Department, Faculty of Science, Taibah University, Saudi Arabia
3 Astronomy Department, National Research Institute of Astronomy and Geophysics, Cairo,
Egypt
4 Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt

Abstract. An edge irregular total k-labeling on simple and undirected graph G(V,E) is a map f :
V ∪ E → {1, 2, . . . , k} such that for any different edge xy and x

′
y

′
their weights f(x) + f(xy) +

f(y) and f
(
x

′
)

+ f
(
x

′
y

′
)

+ f(y
′
) are distinct. The minimum positive integer k for which the

graph G has an edge irregular total k-labeling is called the total edge irregularity strength of G and
is denoted by tes(G). In different fields in our life, like physics, coding theory and computer science,
graph labeling plays a vital role and appears in many applications. A labeling of a graph M(V,E)
is a map which assigns each element in G with a positive integer number. An edge irregular total ζ
-labeling is a function Ω : V (M)∪E(M) → {1, 2, 3, ...., ζ} such that WΩ(h) ̸= WΩ(z) where WΩ(h)
and WΩ(z) are weights for any two distinct edges. In this case, M has total edge irregularity
strength (TEIS) if ζ is minimum. In our paper, we defined a new type of graphs called a triple
star snake graph PS3,nand m-star snake graph PSm,n. Also, we investigated TEIS for a triple star
snake graph PS3,n . We then generalized the results for m-star snake graph PSm,n.

Key Words and Phrases: Edge labeling, Irregularity strength, Irregular labelling Total edge
irregularity strength, Star snake graph

1. Introduction

In real-world systems, interactions between pairs of entities take place every day. Ex-
amples of these systems include human interactions, financial networks, social networks,
and biological networks. In the field of graph theory, such pairs of entities are referred to
as a network, where the substances represent the vertices and the connections between any
two substances are denoted as edges [1, 2]. The use of graph theory in condensed matter
physics, pioneered by the work of many chemical and physical g(Harary, 1968; Trinajstić,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5673

Email addresses: Hala.attiya@Techedu.bsu.edu.eg (H. Attiya),
nkhalifa@taibahu.edu.sa (N. Ahmed), fatma.salama@science.tanta.edu.eg (F. Salama)

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



H. Attiya, N. Ahmed, F. Salama / Eur. J. Pure Appl. Math, 18 (2) (2025), 5673 2 of 11

1992), is well stablished and gaining more popularity. Some of the most important ar-
eas of application of graph theory in physics include condensed matter physics, statistical
physics, quantum electrodynamics, electrical networks and vibrational problems [3–5].

The graph G is the ordered pair (V (G), E(G)) where V (G) is the set of elements called
vertices and E(G) is a finite set of pairs of distinct elements V (G) called set of edge. Two
vertices such as v1 and v2 are called adjacent, whenever v1, v2 ∈ E(G) . Labeling of graph
is a map that carries graph elements to positive integers [6]. The domain of mapping is a
vertex set, or an edge set, or a union of vertex and edge sets. If the domain is a vertex
set, the labeling is called vertex labeling. If the domain is an edge set, the labeling is
called edge labeling. If the domain is a union of vertex and edge sets, the labeling is called
total labeling. On progress, several types of labeling that has been studied can be seen on
Gallian [7].

Spectral graph theory is a beautiful branch of graph theory that utilizes the eigenval-
ues and eigenvectors of matrices naturally associated with graphs to study them. Some
interesting research has been done in the field of spectral graph theory in the past few
years. The random walks of octagonal cell network has been investigated in [8] using the
Laplacian spectrum method where the mean first passage time (τ) and Kemeny’s constant
Ξ between nodes was obtained. The work also provide an explicit expression of Kemeny’s
constant and mean first passage time for octagonal cell network, by their Laplacian eigen-
values and the correlation among roots of characteristic polynomial. In [1], an explicit
closed-form formula of the global meanfirst-passage time (GMFPT) for hexagonal model
has been established using the decomposition theorem of Laplacian polynomial and char-
acteristic polynomial. They have also shown that, extensive matrix analysis, obtaining
GMFPT via spectrums provides an easy calculation in terms of large networks. In [9], the
electric network approach and the combinatorial approach have been used to derive the
exact expression for resistance distances between any two vertices of the kn4 ring model.
The mean first passage time and Kemeny constant of kn4 have also been calculated. A
study of mean-first-passage time and Kemeny’s constant of a random walk by normal-
ized Laplacian matrices of a penta-chain network has been prformed in [10]. Motivtaed
by many applications in computer networks, routing protocols, wireless sensor networks,
and also by the normalized Laplacian (NL) matrix, the spectrums of the n copies of (kl5)
chain graph has been obtained in [11]. The number of spanning trees for kl5 has also been
calculated by utilizing these spectrums.

For a connected and simple graph M(V,E), an edge irregular total ζ-labeling has been
introduced by Baca et al. in [12] as a map Ω : V (M) ∪ E(M) → {1, 2, 3, ...., ζ} such that
WΩ(h) ̸= WΩ(z) where WΩ(h) and WΩ(z) are weights for any two distinct edges. Also,
the inequality of TEIS for a graph, with the maximum degree of vertices ∆G, has been
deduced in the form

tes(M) ≥ max

{
E(M) + 2

3
,

∆(M) + 1

2

}
(1)

Since then, many authors have begun to find TEIS for many families of graphs. Ivanĉo



H. Attiya, N. Ahmed, F. Salama / Eur. J. Pure Appl. Math, 18 (2) (2025), 5673 3 of 11

and Jendrôı in [13] determined TEIS for a tree as

tes(T ) = max

{
k + 2

3
,

∆(M) + 1

2

}
(2)

Ahmad et al. [14–20] have investigated TEIS for zigzag graphs, helm and sun graphs, the
categorical product of two cycles, the categorical product of two paths, the generalized
Petersen graph, certain families of graphs and some classes of plane graphs. Therefore,
TEIS has been determined for hexagonal grid graphs in Al-Mushayt and Ahmad [21],
planar graphs in Yang et al. [22], for some classes of plane graphs in Tarawneh et al. [23],
for fan, wheel, triangular book, and friendship graphs in Tilukay et al. [24], for subdivision
of star in Siddiqui [25], for some Cartesian product graphs in Ramdan and Salman [26], for
trees in Amar and Togn [27], for generalized web graphs and related graphs in Indriat et
al. [28], for generalized prism in Bača and Siddiqui [29], for complete graph and complete
bipartite graphs in Jendrôı et al. [30], for the disjoint union of wheel graphs in Jeyanth
and Sudhai [31], for dense graphs in Majersk et al. [32], for the grids in Mǐskuf and Jendrôı
[33], for disjoint union of isomorphic copies of generalized Petersen graph in Naeem and
Siddiqui [34], for large graphs in Pfender [35], for centralized uniform theta graphs in
Putra and Susanti [36], for series parallel graphs in Rajasingh et al. [37].

Salama [38],[39],[40],[41],[42] has determined TEIS for the polar grid graph, special
families of graphs, heptagonal snake graph, uniform theta snake graphs and quintet snake
graph.

In this paper, we define new types of graphs called a triple star snake graph PS3,n and
m-star snake graph PSm,n. Also, we investigate the TEIS for a triple star snake graph
S3,n. Then, we generalize the results for m-star snake graph PSm,n.

2. Main results

In this section, we define the star snake graph and some related graphs. We also
determine the TEIS for these graphs.

Definition 1. In a path Pn, if we replace every edge with a star S3 we get a new graph
called a triple star snake graph, denoted PS3,n (see Figure 1).

Figure 1: A triple star snake graph PS3,n

Theorem 1. If PS3,n is a triple star snake graph with 3n+1 vertices, then TEIS is given
by : tes(PS3,n) = n + 1



H. Attiya, N. Ahmed, F. Salama / Eur. J. Pure Appl. Math, 18 (2) (2025), 5673 4 of 11

Proof. Since |E(PS3,n)| = 3n and ∆(PS3,n) = 3, Then inequality (1) becomes
tes(PS3,n) ≥ n + 1.
To complete the proof, we will prove the inverse inequality.
Let ζ = n + 1 and Ω : V (PS3,n) ∪ E(PS3,n) → {1, 2, 3, ...., ζ} is a total ζ-labeling defined
as:
Ω(xδ) = δ for δ ∈ {1, 2, 3, ...., n + 1}
Ω(xδyδ) = δ for δ ∈ {1, 2, 3, ...., n}
Ω(yδxδ+1) = Ω(yδzδ) = δ + 1 for δ ∈ {1, 2, 3, ...., n}
The above equations mean that ζ = n + 1 is the greatest label of edges and vertices. The
weights of edges are given by:
WΩ(xδyδ) = 3δ for δ ∈ {1, 2, 3, ...., n}
WΩ(yδxδ+1) = 3δ + 2 for δ ∈ {1, 2, 3, ...., n}
WΩ(yδzδ) = 3δ + 1 for δ ∈ {1, 2, 3, ...., n}
It is clear that the edges weights are dissimilar. Then, tes(PS3,n) ≥ n + 1 .

Definition 2. The m-star snake graph PSm,n is a path Pn in which we replace each edge
with a star Sm (see Fig. 2, 3).

Figure 2: m-star snake graph PSm,n, (m is odd)

Figure 3: m-star snake graph PSm,n, (m is even)

Theorem 2. The TEIS of the m-star snake graph PSm,n with mn + 1 vertices n > 1 is
given by

tes(PSm,n) =
mn + 2

3
(3)



H. Attiya, N. Ahmed, F. Salama / Eur. J. Pure Appl. Math, 18 (2) (2025), 5673 5 of 11

Proof. By substituting with |E(PS4,n)| = mn and ∆(PSm,n) = m in inequality (3),
we find

tes(PSm,n) ≥ mn + 2

3
(4)

An edge irregular total ζ-labeling will be shown assuming that a map Ω : E(PSm,n) ∪
V (PSm,n) → {1, 2, 3, ...., ζ} is a total ζ-labeling defined in two cases:
Case 1: m is even, Ω is defined as:

Ω(xδ) =


m
2 (δ − 1) + 1 for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

+ 1
}

ζ for δ ∈
{

ζ−1
m
2

+ 2, ..., n + 1
}

Ω(yδ) =


m
2 (δ − 1) + 1 for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

+ 1
}

ζ for δ ∈
{

ζ−1
m
2

+ 2, ..., n
}

Ω(hsδ) = Ω(zsδ) =



(m2 − 1)(δ − 1) + s for δ ∈
{

1, 2, 3, ..., ζ−1
m
2

+ 1
}

(m2 − 1)(δ − 1) + s for


δ = ζ−1

m
2

+ 2

s =
{

1, 2, 3, ..., ζ − (m2 − 1)(δ − 1)
}

ζ for


δ = ζ−1

m
2

+ 2

s =
{
ζ − (m2 − 1)(δ − 1) + 1, ..., m2 − 1

}
ζ for δ ∈

{
ζ−1
m
2

+ 3, ..., n
}

Ω(xδyδ) =


1 for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

+ 1
}

(δ − 1)m− 2ζ + 3 for δ ∈
{

ζ−1
m
2

+ 2, ..., n
}

Ω(yδxδ+1) =


m
2 for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

}
m
2 (δ + 1) − ζ + 1 for δ = ζ−1

m
2

+ 1

δm− 2ζ + 2 for δ ∈
{

ζ−1
m
2

+ 2, ..., n
}



H. Attiya, N. Ahmed, F. Salama / Eur. J. Pure Appl. Math, 18 (2) (2025), 5673 6 of 11

Ω(yδz
s
δ) =



δ + s for δ ∈
{

1, 2, 3, ..., ζ−1
m
2

+ 1
}

m
2 (δ − 1) − ζ + δ + s + 1 for


δ = ζ−1

m
2

+ 2

s =
{

1, 2, 3, ..., ζ − (m2 − 1)(δ − 1)
}

(δ − 1)m− 2ζ + 2s + 2 for


δ = ζ−1

m
2

+ 2

s =
{
ζ − (m2 − 1)(δ − 1) + 1, ..., m2 − 1

}
(δ − 1)m− 2ζ + 2s + 2 for δ ∈

{
ζ−1
m
2

+ 3, ..., n
}

Ω(yδh
s
δ) =



δ + s + 1 for δ ∈
{

1, 2, 3, ..., ζ−1
m
2

+ 1
}

m
2 (δ − 1) − ζ + δ + s + 2 for


δ = ζ−1

m
2

+ 2

s =
{

1, 2, 3, ..., ζ − (m2 − 1)(δ − 1)
}

(δ − 1)m− 2ζ + 2s + 3 for


δ = ζ−1

m
2

+ 2

s =
{
ζ − (m2 − 1)(δ − 1) + 1, ..., m2 − 1

}
(δ − 1)m− 2ζ + 2s + 3 for δ ∈

{
ζ−1
m
2

+ 3, ..., n
}

From the previous equations, we can say ζ is the maximum number which labels
vertices and edges. The edges’ weights of PSm,n are given by:

WΩ(xδyδ) = (δ − 1)m + 3

WΩ(yδxδ+1) = mδ + 2

WΩ(yδz
s
δ) = (δ − 1)m + 2(s + 1)

WΩ(yδh
s
δ) = (δ − 1)m + 2s + 3

We can say that from the previous equations, the weights are distinct for any two edges.
So

tes(PSm,n) =
mn + 2

3

Case 2: m is odd, Ω is defined as:

Ω(xδ) =


m
2 (δ − 1) + 1 for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

+ 1
}

ζ for δ ∈
{

ζ−1
m
2

+ 2, ..., n
}

Ω(yδ) =


m
2 (δ − 1) + 1 for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

+ 1
}

ζ for δ ∈
{

ζ−1
m
2

+ 2, ..., n + 1
}



H. Attiya, N. Ahmed, F. Salama / Eur. J. Pure Appl. Math, 18 (2) (2025), 5673 7 of 11

Ω(zsδ) =



m
2 (δ − 1) + s for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

}
m
2 (δ − 1) + s for


δ = ζ−1

m
2

+ 1

s =
{

1, 2, 3, ..., ζ − m
2 (δ − 1)

}
ζ for


δ = ζ−1

m
2

+ 1

s =
{
ζ − m

2 (δ − 1) + 1, ..., m2
}

ζ for δ ∈
{

ζ−1
m
2

+ 2, ..., n
}

Ω(hsδ) =



m
2 (δ − 1) + s for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

}
m
2 (δ − 1) + s for


δ = ζ−1

m
2

+ 1

s =
{

1, 2, 3, ..., ζ − m
2 (δ − 1)

}
ζ for


δ = ζ−1

m
2

+ 1

s =
{
ζ − m

2 (δ − 1), ..., m2
}

ζ for δ ∈
{

ζ−1
m
2

+ 2, ..., n
}

Ω(xδyδ) =


δ for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

+ 1
}

(δ − 1)m− 2ζ + 3 for δ ∈
{

ζ−1
m
2

+ 2, ..., n
}

Ω(yδxδ+1) =


m
2 + δ for δ ∈

{
1, 2, 3, ..., ζ−1

m
2

}
mδ − ζ + 1 − m

2 (δ − 1) for δ = ζ−1
m
2

+ 1

δm− 2ζ + 2 for δ ∈
{

ζ−1
m
2

+ 2, ..., n
}

Ω(yδz
s
δ) =



δ + s for δ ∈
{

1, 2, 3, ..., ζ−1
m
2

}
δ + s for


δ = ζ−1

m
2

+ 1

s =
{

1, 2, 3, ..., ζ − (m2 − 1)(δ − 1)
}

m
2 (δ − 1) − ζ + 2s + δ for


δ = ζ−1

m
2

+ 1

s =
{
ζ − m

2 (δ − 1) + 1, ..., m2
}

(δ − 1)m− 2ζ + 2s + 2 for δ ∈
{

ζ−1
m
2

+ 2, ..., n
}



H. Attiya, N. Ahmed, F. Salama / Eur. J. Pure Appl. Math, 18 (2) (2025), 5673 8 of 11

Ω(yδh
s
δ) =



δ + s + 1 for δ ∈
{

1, 2, 3, ..., ζ−1
m
2

}
δ + s + 1 for


δ = ζ−1

m
2

+ 1

s =
{

1, 2, 3, ..., ζ − m
2 (δ − 1) − 1

}
m
2 (δ − 1) − ζ + 2s + δ + 1 for


δ = ζ−1

m
2

+ 1

s =
{
ζ − (m2 − 1)(δ − 1), ..., m2 − 1

}
(δ − 1)m− 2ζ + 2s + 3 for δ ∈

{
ζ−1
m
2

+ 2, ..., n
}

From the previous formulas, we can deduce that ζ is the greatest label of edges and
vertices. After calculating the weights of the edges of the graph PSm,n we find:

WΩ(xδyδ) = (δ − 1)m + 3

WΩ(yδxδ+1) = mδ + 2

WΩ(yδz
s
δ) = (δ − 1)m + 2(s + 1)

WΩ(yδh
s
δ) = (δ − 1)m + 2s + 3

From the equations of weights of edges we see that they are different. So Ω is an edge
irregular total ζ-labeling and

tes(PSm,n) =
mn + 2

3

Figure 4: 10-star snake graph PS10,5

3. Conclusion

Graph labeling plays an important role in different research areas such as computer
science, coding and mathematical physics. The total edge irregularity strength of a graph
G and is denoted by tes(G) and is defined as the minimum positive integer k for which
the graph G has an edge irregular total k-labeling. The present work aims to study the
star snake graph and some related graphs, and determine the TEIS for these graphs. New



H. Attiya, N. Ahmed, F. Salama / Eur. J. Pure Appl. Math, 18 (2) (2025), 5673 9 of 11

Figure 5: 9-star snake graph PS9,5

types of graphs called a triple star snake graph PSm,n and m-star snake graph PSm,n were
defined . The following theorem has been proved : If PSm,n is a triple star snake graph
with 3n + 1 vertices, then TEIS is : tes(PS3,n) = n + 1. After that, we have generalized
the results for m-star snake graph PSm,nas : tes(PSm,n) = mn+2

3 where the m-star snake
graph PSm,n is defined as a path Pn in which we replace each edge with a star Sm.

Acknowledgements

We are so grateful to the reviewer for his/her many valuable suggestions and comments
that significantly improved the paper.

Conflict of interest All authors declare no conflict of interest in this paper.

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