EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 5974
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Density of Some Graphs and their Mycielski Graph

Racma L. Sango1,∗, Isagani S. Cabahug, Jr.1

1 Department of Mathematics, College of Arts and Sciences, Central Mindanao University,
University Town, Musuan, 8710 Maramag, Bukidnon, Philippines

Abstract. Graph density offers an immediate measure of how tightly a network is connected. This
paper gives explicit closed-form formulas for the densities of nine well-known undirected graph,
simple, and connected graph families: barbell, friendship, sunlet, banana, lollipop, gear, tadpole,
wheel, and fan graphs and their corresponding Mycielski graph. The result supply ready to use
expressions that remove the need to build the graphs just to calculate its density. All proofs rely
only on counting vertices and edges and substituting them into the definition of density.

2020 Mathematics Subject Classifications: : 05C07, 05C12

Key Words and Phrases: graph density, Mycielski graph, special graph families

1. Introduction

Graphs model pairwise relationship in diverse area such as sociology, biology, and com-
puter science. A fundamental characteristic of any graph is its density which ranges from
0 for an empty graph to 1 for a complete graph. Density influences algorithmic complexity,
highlights communication bottlenecks, and guides community-detection heuristics. For ex-
ample, in social networks, density measures the degree of interaction or connection within
a community or group, indicating whether relationships are sparse or highly interconnected
[1].

The concept of graph density serves as a fundamental metric in network analysis. A
substantial body of research underscores its significance in both theoretical and applied
contexts, ranging from sociology, computer science, and data mining to bioinformatics and
communication systems. A dense graph often suggests a highly collaborative or interactive
environment, whereas a sparse graph may indicate isolated vertices or limited connectiv-
ity. A newly introduced parameter, referred to in [2] , has recently been utilized to study
the density of various graphs, including corona graphs. In addition to density, centrality
measures are key tools used to analyze the relative importance of nodes within a graph.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.5974

Email addresses: racmasango@gmail.com (R. Sango),
isaganicabahugjr@cmu.edu.ph (I. Cabahug, Jr.)

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



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 2 of 16

These include degree centrality, which counts the number of direct connections a node
has; betweenness centrality, which quantifies how often a node appears on the shortest
paths between other nodes; closeness centrality, which assesses a node’s average distance
to all other nodes in the network; and eigenvector centrality, which considers not just the
number of a node’s connections, but also the importance of the nodes it is connected to.
The concept of centralities and measures has also been examined in other works, such
as [3], [4] and [5]. These metrics, when used in conjunction with graph density, provide
a more comprehensive understanding of network structure, influence, and flow dynam-
ics. Moreover, graph density plays a central role in the study of large-scale and complex
networks. In biological systems, for instance, analyzing protein-protein interaction net-
works with respect to their density can reveal important functional modules or pathways.
In transportation and communication networks, understanding how dense certain regions
are can influence optimization, reliability, and infrastructure planning. In cybersecurity,
analyzing the density of connections can help detect anomalous patterns or vulnerabilities.

By evaluating a network’s density, researchers are better equipped to understand and
visualize its underlying structure, helping to identify trends, cluster formations, com-
munity detection, bottlenecks, and other critical attributes. Density also influences the
choice of algorithms for tasks such as traversal, shortest path computation, and centrality
measures, and is closely linked to concepts such as graph sparsification and complexity
analysis.

This article focuses on nine classical graph that routinely appear in external, labeling,
and network analysis problems: barbell, friendship, sunlet, banana, lollipop, gear, tadpole,
wheel, and fan graph. For each family we determined closed form expression for |V (G)|
and |E(G)|, substituting the into D(G), and simplify to obtain an exact formula. Also,
their respective Mycielski graphs which are of particular interest due to their ability to
increase chromatic number without introducing new triangles are also examined to explore
how density evolves under graph transformations.

Future sections of this paper will explore formal definitions of graph density, methods
of computation, and comparative results based on the selected graph families.

2. Terminology and Notation

A graph G is a finite nonempty set V of objects called vertices together with a possibly
empty set E of 2-element sets of V called edges. To indicate that a graph G has vertex
set V and edge set E, we write G = (V,E). To emphasize that V and E are the vertex set
and edge set of a graph G, we often write V as V (G) and E as E(G). Each edge {u, v}
of G is usually denoted by uv or vu. The number of vertices in a graph G is the order
of G and the number of edges is the size of G. The degree of a vertex v is denoted by
deg(v) and the minimum degree of G is denoted by δ(G) and the maximum degree of G is
denoted by ∆(G) [6]. An empty graph of order n is graph with n vertices where in every
pair of distinct vertices are not adjacent [6].

For an integer n ≥ 1, the path graph Pn is a graph of order n and size n−1 whose ver-
tices can be labeled as v0, v1, v2, ..., vn−1 and whose edges are vi, vi+1 for i = 1, 2, 3, ..., n−2



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 3 of 16

[6]. For an integer n ≥ 1, the path graph Cn is a graph of order n and size n whose vertices
can be labeled as v0, v1, v2, ..., vn−1 and whose edges are vi, vi+1 for i = 1, 2, 3, ..., n− 2 [6].
For n ≥ 3, the fan graph Fn of order n+1 is a graph obtained by connecting a new vertex
v to each vertex of the path Pn [7]. tadpole graph (Tm,n) is defined as a graph obtained
by combining a vertex of cycle Cm with one of leaf of path Pn [8].

A complete graphof order n ≥ 2, denoted by Kn, is a graph with n vertices where in
every pair of distinct vertices are adjacent [6]. Barbell graph Bn is a graph obtained by
connecting two complete graph Kn by an edge [8]. The (m,n)−lollipop graph denoted
by Lm,n is a graph obtained by joining a complete graph Km to a path graph Pn with a
bridge [9].

The friendship graph, denoted by Frn is a set of n triangle having a common vertex
[10]. The sunlet graph Sn is the graph on 2n vertices obtaining by attaching n pendant
edges to a cycle graph Cn [11]. A banana tree graph Bm,n is a graph of order mn + 1
obtained by connecting one leaf on each of m copies of star K1,n−1 with a single root
vertex v that is distinct from all stars [12].

For n ≥ 3, the wheel graph, Wn of order n+ 1 is a graph produced from the complete
product of an isolated vertex and a cycle Cn [7]. A gear graph, denoted by Gn, is obtained
from the wheel graph by adding a vertex between every pair of adjacent vertices of the
cycle [10].

Consider a graph G with V (G) = v1, v2, v3, ..., vn. Apply the following steps to the
graph G:

(i) Take the set of new vertices U = u1, u2, u3, ..., un and add edges from each vertex ui
of U to the vertices vj if the corresponding vertex vi is adjacent to vj in G.

(ii) Take another new vertex w0 and add edges joining each element in U .

Here, the new graph obtained is the Mycielski graph, denoted by µ(G) of graph G [13].
Let G be undirected graph with |V (G)| and |E(G)|. The density of the graph G,

denoted as D(G), is defined as the ratio of the number of edges in the graph to the
maximum possible number of edges between the number of vertices [14].
For the undirected graph, the density is given by the formula:

D(G) =
2|E(G)|

|V (G)|(|V (G)| − 1)
.

Consider Figure 1, it shows that the number of |E(G)| = 12 and |V (G)| = 11. clearly,

D(G) =
2|E(G)|

|V (G)|(|V (G)| − 1)
=

2|12|
11(11− 1)

=
24

11(10)
=

26

110
= 0.218

3. Results

Theorem 1. Let G be a barbell graph (Bn) where n ≥ 3. Then

D(Bn) =
n2 − n+ 1

2n2 − n
.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 4 of 16

8

4

2

1

3

5

96 7 10 11
Figure 1: The density of D(G) = 0.218

Proof. A barbell graph Bn is constructed by taking two disjoint complete graph Kn

and adding exactly one edge that connects one vertex first in Kn1 to one vertex in the
second Kn2 . Thus the number of edges of the barbell graph Bn is:

|E(Bn)| = 2

(
n(n− 1)

2

)
+ 1 = n(n− 1) + 1 = n2 − n+ 1.

and the number of vertices of the barbell graph Bn is |V (Bn)| = 2n. Therefore,

G(Bn) =
2|E(Bn)|

|V (Bn)|(|V (Bn)| − 1)
=

2(n2 − n+ 1)

2n(2n− 1)
=

n2 − n+ 1

2n2 − n
.

Illustration: Consider Figure 2, it shows that the number of edges and vertices of B4 are
|E| = 13 and |V | = 8 respectively. Clearly,

D(B5) =
2|E(B5)|

|V (B5)|(|V (B5)| − 1)

2|13|
8(8− 1)

=
26

8(7)
=

26

56
= 0.4642

or

D(B5) =
n2 − n+ 1

2n2 − n
=

2(42 − 2(4) + 2)

4(42)− 2(4)
=

26

56
= 0.4642.

Figure 2: The Barbell Graph B4

Theorem 2. Let G be a friendship graph (Frn) where n ≥ 2. Then

D(Frn) =
3n

2n2 + n
.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 5 of 16

Proof. a friendship graph Frn is constructed by n triangles with a common vertex.
Thus the number of edges and vertices of Frn are |E(Frn)| = 3n and |V (Frn)| = 3n− 2
respectively. Therefore,

D(Frn) =
2|E(Frn)|

|V (Frn)|(|V (Frn)| − 1)
=

2(3n)

(2n+ 1)(2n+ 1− 1)
=

6n

(2n+ 1)(2n)

=
3n

(2n+ 1)n
=

3n

2n2 + n
.

Illustration: Consider Figure 3, it shows that the number of edges and vertices of Frn are
|E(Fr3)| = 9 and |V (Fr3)| = 7 respectively. Clearly,

D(F3) =
2|E(Fr3)|

|V (Fr3)|(|V (Fr3)| − 1)
=

18

7(6)
=

18

42
= 0.429

or

D(F3) =
3n

2n2 + n
=

3(3)

3(3)2 − 5(3) + 2
=

9

9− 15 + 2
= 0.429.

v1

v2 v3

v4

v5v6

v0

Figure 3: The Friendship Graph Fr3

Theorem 3. Let G be a sunlet graph (Sn) where n ≥ 3. Then

D(Sn) =
2

2n− 1
.

Proof. A sunlet graph Sn is constructed by adding n pendant edges to a cycle graph Cn.
Thus, the number of edges and vertices of the sunlet graph Sn is |E(Sn)| = |V (Sn)| = 2n.
Therefore,

D(Sn) =
2|E(Sn)|

|V (Sn)|(|V (Sn)| − 1)
=

2(2n)

2n(2n− 1)
=

2

2n− 1
.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 6 of 16

Illustration: Consider Figure 4, it shows that the number of edges and vertices of S4 are
|E(S4)| = 8 and |V (S4)| = 8 respectively. Clearly,

D(S4) =
2|E(S4)|

|V (S4)|(|V (S4)| − 1)
=

2|8|
8(7)

=
16

56
=

2

7
= 0.2857

or

D(S4) =
2

2n− 1
=

2

2(4) + 1
=

2

7
= 0.2857.

v1 v2

v3v4

v′1 v′2

v′4 v′3

Figure 4: The Sunlet Graph S4

Theorem 4. Let G be a banana graph Bn,k where n ≥ 2 and k ≥ 4. Then

D(Bn,k) =
2

nk + 1
.

Proof. A banana graph Bn,k is constructed by connecting one leaf of each n copies of
k-star graph with one root vertex. Thus, the number of edges and vertices of the banana
graph Bn,k are |E(Bn,k)| = nk and |V (Bn,k)| = nk + 1 respectively. Therefore,

D(Bn,k) =
2|E(Bn,k)|

|V (Bn,k)|(|V (Bn,k)| − 1)
=

2(nk)

nk + 1(nk + 1− 1)
=

2nk

(nk + 1)nk
=

2

nk + 1
.

Illustration: Consider Figure 5, it shows that the number of edges and vertices of B3,4 are
|E(B3,4)| = 12 and |V (B3,4)| = 13 respectively. Clearly,

D(B3,4) =
2|E(B3,4)|

|V (B3,4)|(|V (B3,4)| − 1)
=

2|12|
13(13− 1)

=
24

13(12)
=

24

156
= 0.153

or

D(B3,4) =
2

nk + 1
=

2

3(4) + 1
=

2

13
= 0.153.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 7 of 16

v0

v1 v2

v3

v6

v4

v5

v7

v9

v8

v10

v11

v
Figure 5: The Banana Graph B3,4

Theorem 5. Let G be a lollipop graph (Lm,n) where m ≥ 3 and n ≥ 1. Then

D(Lm,n) =
m(m− 1) + 2n

(m+ n)(m+ n− 1)
.

Proof. A lollipop graph Lm,n is constructed by a complete graph Kn and a path graph
Pn adding exactly one edge that connect one vertex first in Kn to one vertex in Pn. Thus,
the number of edges and vertices of the lollipop graph Lm,n are

|E(Lm,n)| =
m(m− 1) + 2n

2
and |V (Lm,n)| = m+ n respectively. Therefore,

D(Lm,n) =
2|E(Lm,n)|

|V (Lm,n)|(|V (Lm,n)| − 1)
=

2

[
m(m− 1) + 2n

2

]
(m+ n)(m+ n+ 1)

=
m(m− 1) + 2n

(m+ n)(m+ n+ 1)
.

Illustration: Consider Figure 6, it shows that the number edges and vertices of L4,3 are
|E(L4,3)| = 9 and |V (L4,3)| = 7. Clearly,

D(L4,3) =
2|E(L4,3)|

|V (L4,3)|(|V (L4,3)| − 1)

2|9|
7(7− 1)

=
18

7(6)
=

18

42
= 0.43

or

D(L4,3) =
m(m− 1) + 2n

(m+ n)(m+ n+ 1)
=

4(4− 1) + 2(3)

(4 + 3)(4 + 3− 1)
=

18

42
= 0.43.

v1

v2

v3

v4 v5 v6 v7

Figure 6: The Lollipop Graph L4,3



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 8 of 16

Theorem 6. Let G be a tadpole graph (Tm,n) where m ≥ 3 and n ≥ 1. Then

D(Tm,n) =
2

m+ n− 1
.

Proof. A tadpole graph Tm,n is constructed by cycle graph Cn and path graph Pn

adding exactly one edge that connect one vertex in Cn to one vertex in Pn. Thus, the
number of vertices and edges of the tadpole graph Tm,n is |E(Tm,n)| = |V (Tm,n)| = m+n.
Therefore,

D(Tm,n) =
2|E(Tm,n)|

|V (Tm,n)|(|V (Tm,n)| − 1)
=

2(m+ n)

(m+ n)(m+ n− 1)
=

2

m+ n− 1
.

Consider Figure 7, it shows that the number edges and vertices of T4,3 are |E(T4,3)| = 7
and |V (T4,3)| = 7 respectively. Clearly,

D(T4,3) =
2|E(T4,3)|

|V (T4,3)|(|V (T4,3)| − 1)
=

2|7|
7(7− 1)

=
14

7(6)
=

14

42
= 0.33

or

D(L4,3) =
2

m+ n− 1
=

2

(4 + 3− 1))
=

2

6
= 0.33.

Figure 7: The Tadpole Graph T4,3

Theorem 7. Let G be a gear graph Gn where n ≥ 3. Then

D(Gn) =
3

2n+ 1
.

Proof. A gear graph is constructed by wheel graph Wn adding vertex in every pair of
adjacent in the outer cycle. Thus, the number of edges and vertices of the gear graph Gn

are |E(Gn)| = 3n and the order is |V (Gn)| = 2n+ 1 respectively. Therefore,

D(Gn) =
2|E(Gn)|

|V (Gn)|(|V (Gn)| − 1)
=

2(3n)

(2n+ 1)(2n+ 1− 1)
=

2(3n)

(2n+ 1)2n
=

3

2n+ 1
.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 9 of 16

Illustration: Consider Figure 8, it shows that the number of edges and vertices of G4 are
|E(G4)| = 16 and |V (G4)| = 9 respectively. Clearly,

D(G4) =
2|E(G4)|

|V (G4)|(|V (G4)| − 1)
=

2|16|
9(9− 1)

=
32

9(8)
=

32

72
= 0.44

or

D(G4) =
3

2n+ 1
=

4

2(4) + 1
=

4

9
= 0.44.

v1

v2

v3

v4

c

b a

d

v0

Figure 8: The Gear Graph G4

Theorem 8. Let G be a wheel graph (Wn) where n ≥ 4. Then

D(Wn) =
4

n
.

Proof. A wheel graph Wn is constructed by cycle graph Cn−1 and consists one central
vertex that connects to all vertices in Cn−1. Thus, the number of edges and vertices of
the wheel graph are |E(Wn)| = 2n and order is |V (Wn)| = n+ 1. Therefore,

D(Wn) =
2|E(Wn)|

|V (Wn)|(|V (Wn)| − 1)
=

2(2n)

(n+ 1)(n+ 1− 1)
=

4n

(n+ 1)n
=

4

n+ 1
.

Illustration: Consider Figure 9, it shows that the number edges and vertices of W5 are
|E(W5)| = 10 and |V (W5)| = 6 respectively. Clearly,

D(W5) =
2|E(W5)|

|V (W5)|(|V (W5)| − 1)
=

2|10|
6(6− 1)

=
20

6(5)
=

20

30
= 0.67

or

D(W5) =
4

n+ 1
=

4

5 + 1
=

4

6
= 0.67.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 10 of 16

v1

v

v3

v5 v4

v2

W6 :

Figure 9: The Graph W5

Theorem 9. Let G be a fan graph graph (Fn) where n ≥ 3. Then

D(Fn) =
4n− 2

(n+ 1)(n)
.

Proof. A fan graph Fn is constructed by a path graph Pn and one central vertex that
connects to all vertices in path graph Pn. Thus, the number of edges and vertices of the
fan graph Fn are |E(Fn)| = 2n− 1 and order |V (Fn)| = n+ 1 respectively. Therefore,

D(Fn) =
2|E(Fn)|

|V (Fn)|(|V (Fn)| − 1)
=

2m

(n+ 1)(n+ 1− 1)
=

2(2n− 1)

n(n− 1)
=

4n− 2

(n+ 1)n
.

Illustration: Consider Figure 10, it shows that the number of edges and vertices of F5 are
|E(F5)| = 9 and |V (F5)| = 6. Clearly,

D(F5) =
2|E(F5)|

|V (F5)|(|V (F5)| − 1)
=

2|9|
6(6− 1)

=
18

6(5)
=

18

30
= 0.6

or

D(F5) =
4n− 2

(n+ 1)n
=

4(5)− 2

(5 + 1)5
=

18

(5)5
=

18

28
= 0.6.

1

2

3

4

5

v
Figure 10: The Fan Graph F5



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 11 of 16

By Definition Mycielski Graph, we can easily derived the number of edges and vertices
of a Mycielski graph of a connected graph.

Remark 1. Let G be a nontrivial connected graph. Then |E(µ(G))| = 3|E(G)|+ |V (G)|
and |V (µ(G))| = 2|V (G)|+ 1.

Theorem 10. Let G be a nontrivial connected graph. Then the density of µ(G) is

D(µ(G)) =
3|E(G)|+ |V (G)|

|V (G)|(2|V (G)|+ 1)
.

Proof. By Definition of Mycielski graph and Remark 1,

D(µ(G)) =
2|E(µ(G))|

|V (G)|(|V (G)| − 1)
=

2(3|E(G)|+ |V (G)|)
(2|V (G)|+ 1)(2|V (G)|+ 1− 1)

=
2(3|E(G)|+ |V (G)|)
(2|V (G)|+ 1)2|V (G)|

=
3|E(G)|+ |V (G)|

|V (G)|(2|V (G)|+ 1)
.

Corollary 1. Let G be a path graph (Pn) where n ≥ 2. Then,

D(µ(Pn)) =
4n− 3

2n2 + n
.

Proof. By Theorem 10 and since the number of edges and vertices of Pn are n− 1 and
n respectively, we have

D(µ(Pn)) =
3|E(Pn)|+ |V (Pn)|

|V (Pn)|(2|V (Pn)|+ 1)
=

3(n− 1) + n

n(2n+ 1)
=

3n− 3 + n

2n2 + n
=

4n− 3

2n2 + n
.

Corollary 2. Let G be a cycle graph (Cn) with n ≥ 3. Then

D(µ(Cn)) =
4

2n+ 1
.

Proof. By Theorem 10 and since the number of edges and vertices of Cn is n, we have

D(µ(Cn)) =
3|E(Cn)|+ |V (Cn)|

|V (Cn)|(2|V (Cn)|+ 1)
=

3n+ n

n(2n+ 1)
=

4n

n(2n+ 1)
=

4

2n+ 1
.

Corollary 3. Let G be a complete graph (Kn) where n ≥ 4. Then

D(µ(Kn)) =
3n− 1

4n+ 2
.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 12 of 16

Proof. By Theorem 10 and since the number of edges and vertices of Kn are
n(n− 1)

2
and n respectively, we have

D(µ(Kn)) =
3|E(Kn)|+ |V (Kn)|

|V (Kn)|(2|V (Kn)|+ 1)
=

3(n2 − n)

2
+ n

2n2 + n
=

3n2 − 3n+ 2n

2
2n2 + n

=

3n2 − n

2
2n2 + n

=

(
3n2 − n

2

)(
1

2n2 + n

)
=

3n2 − n

4n2 + 2n
=

n(3n− 1)

n(4n+ 2)

=
3n− 1

4n+ 2
.

Corollary 4. Let G be a barbell graph (Bn) where n ≥ 3. Then

D(µ(Bn)) =
3n2 − n+ 3

8n2 + 2n
.

Proof. By Theorem 10 and since the number of edges and vertices of Bn are n2−n+1
and 2n respectively, we have

D(µ(Bn)) =
3|E(Bn)|+ |V (Bn)|

|V (Bn)|(2|V (Bn)|+ 1)
=

3(n2 − n+ 1) + 2n

2n(2 · 2n+ 1)
=

3n2 − 3n+ 3 + 2n

8n2 + 2n

=
3n2 − n+ 3

8n2 + 2n
.

Corollary 5. Let G be a friendship graph (Frn) where n ≥ 2. Then

D(µ(Frn)) =
11n+ 1

8n2 + 10n+ 3
.

Proof. By Theorem 10 and since the number of edges and vertices of Frn are 3n and
3n− 2 respectively, we have

D(µ(Frn)) =
3|E(Frn)|+ |V (Frn)|

|V (Frn)|(2|V (Frn)|+ 1)
=

3(3n) + 2n+ 1

(2n+ 1)[2(2n+ 1) + 1]
=

9n+ 2n+ 1

(2n+ 1)(4n+ 3)

=
11n+ 1

8n2 + 10n+ 3
.

Corollary 6. Let G be a sunlet graph (Sn) where n ≥ 3. Then

D(µ(Sn)) =
4

4n+ 1
.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 13 of 16

Proof. By Theorem 10 and since the number of edges and vertices of Sn is 2n, we have

D(µ(Sn)) =
3|E(Sn)|+ |V (Sn)|

|V (Sn)|(2|V (Sn)|+ 1)
=

3(2n) + 2n

2n(2 · 2n+ 1)
=

2n(4)

2n(4n+ 1)
=

4

4n+ 1
.

Corollary 7. Let G be a banana graph (Bn,k) where n ≥ 2 and k ≥ 4. Then

D(µ(Bn,k)) =
4nk + 1

2n2k2 + 5nk + 3
.

Proof. By Theorem 10 and since the number of edges and vertices of Bn,k are nk and
nk + 1 respectively, we have

D(µ(Bn,k)) =
3|E(Bn,k)|+ |V (Bn,k)|

|V (Bn,k)|(2|V (Bn,k)|+ 1)
=

3nk + nk + 1

(nk + 1)[2(nk + 1) + 1]
=

4nk + 1

(nk + 1)(2nk + 3)

=
4nk + 1

2n2k2 + 5nk + 3
.

Corollary 8. Let G be a lollipop graph (Lm,n) where m ≥ 3 and n ≥ 1 . Then

D(µ(Lm,n)) =
3m2 −m+ 8n

4m2 + 8mn+ 4n2 +m+ n
.

Proof. By Theorem 10 and since the number of edges and vertices of Lm,n are
m(m− 1) + 2

2
and m+ n respectively, we have

D(µ(Lm,n)) =
3|E(Lm,n)|+ |V (Lm,n)|

|V (Lm,n)|(2|V (Lm,n)|+ 1)
=

3

(
m(m− 1) + 2n

2

)
+m+ n

(m+ n)[2(m+ n) + 1]

=

3(m2 −m+ 2n)

2
+m+ n

(m+ n)(2m+ 2n+ 1)

=

3m2 − 3m+ 6n+ 2m+ 2n

2
2m2 + 4mn+ 2n2 +m+ n

=

3m2 −m+ 8n

2
2m2 + 4mn+ 2n2 +m+ n

=
3m2 −m+ 8n

4m2 + 8mn+ 4n2 +m+ n
.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 14 of 16

Corollary 9. Let G be a gear graph (Gn) where n ≥ 3. Then

D(µ(Gn)) =
11n+ 1

8n2 + 10n+ 3
.

Proof. By Theorem 10 and since the number of edges and vertices of Gn are 3n and
2n+ 1 respectively, we have

D(µ(Gn)) =
3|E(Gn)|+ |V (Gn)|

|V (Gn)|(2|V (Gn)|+ 1)
=

3(3n) + 2n+ 1

(2n+ 1)[2(2n+ 1) + 1]
=

9n+ 2n+ 1

(2n+ 1)(4n+ 3)

=
11n+ 1

8n2 + 6n+ 4n+ 3
=

11n+ 1

8n2 + 10n+ 3
.

Corollary 10. Let G be a tadpole graph (Tm,n) where m ≥ 3 and n ≥ 1 . Then,

D(µ(Tm,n)) =
4

2m+ 2n+ 1
.

Proof. By Theorem 10 and since the number of edges and vertices of Tm,n is m + n,
we have

D(µ(Tm,n)) =
3|E(Tm,n)|+ |V (Tm,n)|

|V (Tm,n)|(2|V (Tm,n)|+ 1)
=

3(m+ n) +m+ n

(m+ n)[2(m+ n) + 1]
=

4m+ 4n

(m+ n)(2m+ 2n+ 1)

=
4(m+ n)

(m+ n)(2m+ 2n+ 1)
=

4

2m+ 2n+ 1
.

Corollary 11. Let G be a wheel graph (Wn) where n ≥ 3. Then

D(µ(Wn)) =
7n− 6

2n2 + n
.

Proof. By Theorem 10 and since the number of edges and vertices of Wn are 2n − 2
and n respectively, we have

D(µ(Wn)) =
3|E(Wn)|+ |V (Wn)|

|V (Wn)|(2|V (Wn)|+ 1)
=

3(2n− 2) + n

n(2n+ 1)
=

6n− 6 + n

2n2 + n
=

7n− 6

2n2 + n
.

Corollary 12. Let G be a fan graph (Fn) where n ≥ 3. Then

D(µ(Fn)) =
7n− 2

2n2 + 5n+ 3
.



R. Sango, I. Cabahug, Jr / Eur. J. Pure Appl. Math, 18 (3) (2025), 5974 15 of 16

Proof. By Theorem 10 and since the number of edges and vertices of Fn are 2n − 1
and n+ 1 respectively, we have

D(µ(Fn)) =
3|E(Fn)|+ |V (Fn)|

|V (Fn)|(2|V (Fn)|+ 1)
=

3(2n− 1) + n+ 1

(n+ 1)[2(n+ 1) + 1]
=

6n− 3 + n+ 1

(n+ 1)(2n+ 3)

=
7n− 2

2n2 + 5n+ 3
.

Acknowledgements

The authors would like to extend their heartfelt gratitude and appreciation to the
people who are beyond the success of this paper. Also, the authors would like to express
their profound gratitude to the Department of Science and Technology-Science Education
Institute Science and Technology Regional Alliance of Universities for Inclusive National
Development (DOST-SEI STRAND) for their invaluable assistance throughout the re-
search process.

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