Ratio Mathematica                                                                            Volume 44, 2022 

 
 

 

 

Connected Hub Sets and Connected Hub 

Polynomials of the Lollipop Graph 𝑳𝒑,𝟏 
 

T. Angelinshiny1 

T. Anitha Baby2 

 

Abstract 

Let 𝐺 be a graph with vertex set 𝑉(𝐺). The number of vertices in 𝐺 is the order of 𝐺 and 
is denoted by |𝑉(𝐺)|. The connected hub polynomial of G denoted by 𝐻𝑐 (𝐺, 𝑦) is 

defined as 𝐻𝑐 (𝐺, 𝑦) = ∑ ℎ𝑐 (𝐺, 𝑘)𝑦
𝑘|𝑉(𝐺)|

𝑘=𝒽𝑐(𝐺)
  where ℎ𝑐 (𝐺, 𝑘) denotes the number of 

connected hub sets of 𝐺 of cardinality 𝑘 and 𝒽𝑐 (𝐺) denotes the connected hub number 
of 𝐺. Let 𝐿𝑝,1 denotes the Lollipop graph with 𝑝 + 1 vertices. The connected hub 

polynomial of  𝐿𝑝,1 denoted by 𝐻𝑐 (𝐿𝑝,1, 𝑦) is defined as,𝐻𝑐 (𝐿𝑝,1, 𝑦) =

∑ ℎ𝑐 (𝐿𝑝,1, 𝑘)
|𝑉(𝐿𝑝,1)|

𝑘=𝒽𝑐(𝐿𝑝,1)
𝑦𝑘where ℎ𝑐 (𝐿𝑝,1, 𝑘) denotes the number of connected hub sets of 

𝐿𝑝,1 of cardinality 𝑘, and 𝒽𝑐 (𝐿𝑝,1) denotes the connected hub number of 𝐿𝑝,1.In this 

paper, we derive a recursive formula for hc(Lp,1, k). From this recursive formula, we 

construct the connected hub polynomial of Lp,1 as,Hc(Lp,1, y) =

∑ hc(Lp,1, k)
p+1
k=1

ykAlso we study some properties of this polynomial. 

 

Keywords: Lollipop Graph, connected hub set, connected hub number, Connected hub 

polynomial. 

 

Mathematics Subject Classification Code: 05C31, 05C993 

 

 

 

 

 
1Research Scholar (Reg. No. 20213282092009), Department of Mathematics, Women’s Christian 

College, Nagercoil, Tamil Nadu, India. Affiliated by Manonmaniam Sundaranar University, 

Abishekapatti, Tirunelveli - 627 012. Mail Id: angelinshinyt@gmail.com 
2 Assistant Professor, Department of Mathematics, Women’s Christian College, Nagercoil, Tamil Nadu, 

India. Affiliated by Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012. Mail Id: 

anithasteve@gmail.com 
3Received on June 19th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 

10.23755/rm.v44i0.886. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY license agreement. 

29

mailto:angelinshinyt@gmail.com
mailto:anithasteve@gmail.com


T. Angelinshiny and T. Anitha Baby 

1. Introduction 

If any two distinct vertices of a graph 𝐺 are adjacent, then 𝐺 is a complete graph. If 
a tree has two nodes of vertex degree 1 and other nodes of vertex degree 2, then it is a 

path graph. A complete graph of order 𝑝 is denoted by 𝐾𝑝 and a path graph of order 𝑞 is 

denoted by 𝑃𝑞 . Join a complete graph 𝐾𝑝 to a path graph 𝑃𝑞  with a bridge. The resulting 

graph is a Lollipop graph 𝐿𝑝,𝑞. 

 

 

2. Connected Hub Sets of the Lollipop Graph 𝐋𝐩,𝟏 

In this section, we give the connected hub number of the Lollipop graph 𝐿𝑝,1 and 

some of the properties of the connected hub sets of the Lollipop graph 𝐿𝑝,1. 

 

Definition 2.1 Join a complete graph 𝐾𝑝 to a path graph 𝑃1 with a bridge. The resulting 

graph is a Lollipop graph 𝐿𝑝,1. 

 

Definition 2.2 Let 𝐺 = (𝑉, 𝐸) be a connected graph. A subset 𝐻 of 𝑉 is called a hub set 
of 𝐺 if for any two distinct vertices 𝑢, 𝑣 ∈ 𝑉 − 𝐻, there exists a 𝑢 − 𝑣 path 𝑃 in 𝐺, such 
that all the internal vertices of 𝑃 are in 𝐻. The minimum cardinality of a hub set of 𝐺 is 
called the hub number of 𝐺 and is denoted by 𝒽(𝐺). 
 

Definition 2.3 A hub set 𝐻 of 𝐺 is called a connected hub set if the induced subgraph <
𝐻 > is connected. The minimum cardinality of a connected hub set of 𝐺 is called 
connected hub number of 𝐺 and is denoted by 𝒽𝑐 (𝐺). 
 

Theorem 2.4 

ℎ𝑐 (𝐿𝑝,1,𝑘) = {
(

𝑝 + 1
𝑘

) − (
𝑝
𝑘

) + 1 𝑤ℎ𝑒𝑛 𝑘 = 1 𝑎𝑛𝑑 𝑝 − 1                             

(
𝑝 + 1

𝑘
) − (

𝑝
𝑘

)    𝑖𝑓 2 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑  𝑘 ≠   𝑝 − 1          
  

Proof. Let 𝐿𝑝,1 be the Lollipop graph with 𝑝 + 1 vertices and 𝑝 ≥ 4. Let 

𝑣1, 𝑣2, 𝑣3 … 𝑣𝑝, 𝑣𝑝+1  be the vertices of 𝐿𝑝,1, in which the degree of the vertices  

𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑝−1 is 𝑝 − 1, the degree of the vertex 𝑣𝑝 is 𝑝 and the degree of the vertex 

𝑣𝑝+1 is 1. Since, 𝐿𝑝,1 contains 𝑝 + 1 vertices, the number of subsets of 𝐿𝑝,1 with 

cardinality 𝑘 is (
𝑝 + 1

𝑘
). Also, Since, the subgraph with vertex set {𝑣1, 𝑣2, 𝑣3 … 𝑣𝑝−1} is 

not adjacent to 𝑣𝑝+1 every hub set must contain the vertices 𝑣𝑝 or 𝑣𝑝+1. Therefore, 

every time (
𝑝
𝑘

) number of subsets of 𝐿𝑝,1 of cardinality 𝑘 are not connected hub sets.  

Thus, 𝐿𝑝,1 have (
𝑝 + 1

𝑘
) − (

𝑝
𝑘

) number of connected hubs sets of cardinalities 𝑘. 

When the cardinality is 𝑝 − 1, the set which contains 𝑣𝑝−1 is also a connected hub set. 

When the cardinality is 1, {𝑣𝑝} and  {𝑣𝑝+1} are the only connected hub sets. 

30



 Connected Hub Sets and Connected Hub Polynomials of the Lollipop Graph 𝐿𝑝,1 

 

 
 

Hence, ℎ𝑐 (𝐿𝑝,1,𝑘) = {
(

𝑝 + 1
𝑘

) − (
𝑝
𝑘

) + 1 𝑤ℎ𝑒𝑛 𝑘 = 1, 𝑝 − 1                             

(
𝑝 + 1

𝑘
) − (

𝑝
𝑘

)    𝑖𝑓 2 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑  𝑘 ≠   𝑝 − 1          
  

 

Theorem 2.5 Let 𝐿𝑝,1 be the Lollipop graph with 𝑝 ≥ 4.Then 

(i) ℎ𝑐 (𝐿𝑝,1,𝑘) = (
𝑝

𝑘 − 1
) for all 2 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠  𝑝 − 1 

(ii) ℎ𝑐 (𝐿𝑝,1,𝑘) = (
𝑝

𝑘 − 1
) + 1  when 𝑘 = 1 𝑎𝑛𝑑 𝑝 − 1. 

(iii) ℎ𝑐 (𝐿𝑝,1,𝑘) =

{
ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1,𝑘 − 1) 𝑖𝑓 1 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠ 2 , 𝑝 − 2

ℎ𝑐 (𝐿𝑝−1,1, 𝑘) +  ℎ𝑐 (𝐿𝑝−1,1,𝑘 − 1) − 1 𝑖𝑓 𝑘 = 2 𝑎𝑛𝑑 𝑝 − 2                 
 

 

Proof: (i) From Theorem 2.4, we have  

ℎ𝑐 (𝐿𝑝,1,𝑘) = (
𝑝 + 1

𝑘
) − (

𝑝
𝑘

)   

We know that, (
𝑝 + 1

𝑘
) − (

𝑝
𝑘

) = (
𝑝

𝑘 − 1
) 

Therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = (
𝑝

𝑘 − 1
) for all 2 ≤ 𝑘 ≤ 𝑝 + 1 and 𝑘 ≠ 𝑝 − 1.  

(ii) From Theorem 2.4, we have  

ℎ𝑐 (𝐿𝑝,1, 𝑘) = (
𝑝 + 1

𝑘
) − (

𝑝
𝑘

) + 1  𝑤ℎ𝑒𝑛 𝑘 = 1 𝑎𝑛𝑑 𝑝 − 1. 

We know that, (
𝑝 + 1

𝑘
) − (

𝑝
𝑘

) = (
𝑝

𝑘 − 1
) 

Therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = (
𝑝

𝑘 − 1
) + 1 𝑤ℎ𝑒𝑛 𝑘 = 1 𝑎𝑛𝑑 𝑝 − 1. 

(iii) From (i) ℎ𝑐 (𝐿𝑝,1,𝑘) = (
𝑝

𝑘 − 1
) for all 2 ≤ 𝑘 ≤ 𝑝 + 1 and 𝑘 ≠ 𝑝 − 1. 

ℎ𝑐 (𝐿𝑝−1,1, 𝑘) = (
𝑝 − 1
𝑘 − 1

)  

and   ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) = (
𝑝 − 1
𝑘 − 2

)  

Consider, ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) = (
𝑝 − 1
𝑘 − 1

) + (
𝑝 − 1
𝑘 − 2

) = (
𝑝

𝑘 − 1
) =

ℎ𝑐 (𝐿𝑝,1,𝑘)              

Therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) for 1 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠

2, 𝑝 − 2 
When 𝑘 = 2, 

 ℎ𝑐 (𝐿𝑝,1, 2) = (
𝑝
1

)  

ℎ𝑐 (𝐿𝑝−1,1, 2) = (
𝑝 − 1

1
)  

and   ℎ𝑐 (𝐿𝑝−1,1, 1) = (
𝑝 − 1

0
) + 1, by (ii) 

Consider, ℎ𝑐 (𝐿𝑝−1,1, 2) + ℎ𝑐 (𝐿𝑝−1,1, 1) = (
𝑝 − 1

1
) + (

𝑝 − 1
0

) + 1 = (
𝑝
1

) + 1 

31



T. Angelinshiny and T. Anitha Baby 

= ℎ𝑐 (𝐿𝑝,1, 2) + 1 

That is,  ℎ𝑐 (𝐿𝑝,1, 2) = ℎ𝑐 (𝐿𝑝−1,1, 2) + ℎ𝑐 (𝐿𝑝−1,1, 1) − 1.           

Therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) − 1 when 𝑘 = 2. 

When 𝑘 = 𝑝 − 2, 

 ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2) = (
𝑝

𝑝 − 3
)  

ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 2) = (
𝑝 − 1
𝑝 − 3

)  

and   ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 3) = (
𝑝 − 1
𝑝 − 4

) + 1, by (ii) 

Consider, ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 2) + ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 3) = (
𝑝 − 1
𝑝 − 3

) + (
𝑝 − 1
𝑝 − 4

) + 1 =

(
𝑝

𝑝 − 3) + 1 = ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2) + 1 

That is,  ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2) = ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 2) + ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 3) − 1.   

        

Therefore, ℎ𝑐 (𝐿𝑝,1,𝑘) = ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−,1,𝑘 − 1) − 1 when 𝑘 = 𝑝 − 2. 

Hence,  

ℎ𝑐 (𝐿𝑝,1,𝑘) = {
ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1,𝑘 − 1) 𝑖𝑓 1 ≤ 𝑘 ≤ 𝑝 + 1 𝑎𝑛𝑑 𝑘 ≠ 2 , 𝑝 − 2

ℎ𝑐 (𝐿𝑝−1,1, 𝑘) +  ℎ𝑐 (𝐿𝑝−1,1,𝑘 − 1) − 1 𝑖𝑓 𝑘 = 2 𝑎𝑛𝑑 𝑝 − 2                 
 

 

 

3. Connected Hub Polynomials of the Lollipop Graph 

𝑳𝒑,𝟏. 

Definition 3.1 Let 𝐻𝑐 (𝐿𝑝,1,, 𝑘) denotes the family of connected hub sets of the Lollipop 

graph 𝐿𝑝,1, of cardinality 𝑘 and ℎ𝑐 (𝐿𝑝,1,, 𝑘) = |𝐻𝑐 (𝐿𝑝,1,, 𝑘)|. Then, the connected hub 

polynomial of 𝐿𝑝,1 denoted by  𝐻𝑐 (𝐿𝑝,1,, 𝑦) is defined as 𝐻𝑐 (𝐿𝑝,1,, 𝑦) =

∑ ℎ𝑐 (𝐿𝑝,1,, 𝑘)
𝑝+1

𝑘=𝒽𝑐(𝐿𝑝,1,)
𝑦𝑘 

where 𝒽𝑐 (𝐿𝑝,1,) is connected hub number of 𝐿𝑝,1,. 

 

Remark 3.2 𝒽𝑐 (𝐿𝑝,1,) = 1. 

Proof: Label the vertices of 𝐿𝑝,1 by 𝑣1, 𝑣2,𝑣3, … , 𝑣𝑝, 𝑣𝑝+1 in which the degree of the 

vertices  𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑝−1 is 𝑝 − 1, the degree of the vertex 𝑣𝑝 is 𝑝 and the degree of 

the vertex 𝑣𝑝+1 is 1. Since, any two vertices of 𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑝 are adjacent there is a 

path between any two vertices of 𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑝. Also, 𝑣𝑝 is the internal vertex for all 

the path between the vertices of {𝑣1, 𝑣2,𝑣3, … , 𝑣𝑝−1} and 𝑣𝑝+1. Hence {𝑣𝑝} and  {𝑣𝑝+1} 

are two connected hub sets of cardinalities 1.   

Hence,  𝒽𝑐 (𝐿𝑝,1) = 1. 
 

 

32



 Connected Hub Sets and Connected Hub Polynomials of the Lollipop Graph 𝐿𝑝,1 

 

 
 

 

 

Theorem 3.3 𝐻𝑐 (𝐿𝑝,1,𝑦) = (1 + 𝑦)𝐻𝑐 (𝐿𝑝−1,1, 𝑦) − 𝑦
2 − 𝑦𝑝−2 with initial value 

𝐻𝑐 (𝐿4,1,𝑦) = 2𝑦 + 4𝑦
2 + 7𝑦3 + 4𝑦4 + 𝑦5. 

Proof. We have,  𝐻𝑐 (𝐿𝑝,1,𝑦) = ∑ ℎ𝑐 (𝐿𝑝,1, 𝑘)𝑦
𝑘𝑝+1

𝑘=1  

𝐻𝑐 (𝐿𝑝,1,𝑦) = ∑ ℎ𝑐 (𝐿𝑝,1, 𝑘)𝑦
𝑘 +

𝑝+1
𝑘=1          
𝑘≠2,𝑝−2

 ℎ𝑐 (𝐿𝑝,1, 2)𝑦
2 + ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2)𝑦

𝑝−2 

= ∑ [ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1, 𝑘 − 1)]𝑦
𝑘 +

𝑝+1

 
𝑘=1          
𝑘≠2,𝑝−2

[ℎ𝑐 (𝐿𝑝−1,1, 2) +  ℎ𝑐 (𝐿𝑝−1,1, 1)

− 1]𝑦2 +   [ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 2) + ℎ𝑐 (𝐿𝑝−1,1, 𝑝 − 3) − 1]𝑦
𝑝−2 

= ∑ [ℎ𝑐 (𝐿𝑝−1,1, 𝑘) + ℎ𝑐 (𝐿𝑝−1,1, 𝑘 − 1)]𝑦
𝑘 − 𝑦2 − 𝑦𝑝−2

𝑝+1
𝑘=1   

= ∑ ℎ𝑐 (𝐿𝑝−1,1, 𝑘)𝑦
𝑘 + ∑ ℎ𝑐 (𝐿𝑝−1,1, 𝑘 − 1)

𝑝+1
𝑘=1 𝑦

𝑘 − 𝑦2 − 𝑦𝑝−2
𝑝+1
𝑘=1    

= ∑ ℎ𝑐 (𝐿𝑝−1,1, 𝑘)𝑦
𝑘 + 𝑦 ∑ ℎ𝑐 (𝐿𝑝−1,1, 𝑘 − 1)

𝑝+1
𝑘=1 𝑦

𝑘−1 − 𝑦2 − 𝑦𝑝−2
𝑝+1
𝑘=1    

= 𝐻𝑐 (𝐿𝑝−1,1, 𝑦) + 𝑦𝐻𝑐 (𝐿𝑝−1,1, 𝑦) − 𝑦
2 − 𝑦𝑝−2 

= (1 + 𝑦)𝐻𝑐 (𝐿𝑝−1,1, 𝑦) − 𝑦
2 − 𝑦𝑝−2  

Hence, 𝐻𝑐 (𝐿𝑝,1,𝑦) = (1 + 𝑦)𝐻𝑐 (𝐿𝑝−1,1, 𝑦) − 𝑦
2 − 𝑦𝑝−2 with initial value   

𝐻𝑐 (𝐿4,1,𝑥) = 2𝑦 + 4𝑦
2 + 7𝑦3 + 4𝑦4 + 𝑦5. 

 

Example 3.4 Consider the Lollipop graph  𝐿5,1 be with order 6 given in Figure 1. 
 

 
Figure 1 

             

𝐻𝑐 (𝐿5,1,𝑦)  =  2𝑦 + 5𝑦
2 + 10𝑦3 + 11𝑦4 + 5𝑦5 + 𝑦6 

By Theorem 3.3, we have, 

𝐻𝑐 (𝐿5,1,𝑦) = (1 + 𝑦)𝐻𝑐 (𝐿4,1, 𝑦) − 𝑦
2 − 𝑦3         

= (1 + 𝑦)(2𝑦 + 4𝑦2 + 7𝑦3 + 4𝑦4 + 𝑦5) −𝑦2 − 𝑦3 
= 2𝑦 + 5𝑦2 + 10𝑦3 + 11𝑦4 + 5𝑦5 + 𝑦6 
 

Theorem 3.5 Let 𝐿𝑝,1 be the Lollipop graph with 𝑝 ≥ 4.Then 

(𝑖) 𝐻𝑐 (𝐿𝑝,1,𝑦) = ∑ (
𝑝 + 1

𝑘
)

𝑝+1
𝑘=1 𝑦

𝑘 − ∑ (
𝑝
𝑘

)
𝑝+1
𝑘=1 𝑦

𝑘 − 𝑦2 − 𝑦𝑝−2. 

33



T. Angelinshiny and T. Anitha Baby 

(ii) 𝐻𝑐 (𝐿𝑝,1,   𝑦) = ∑ (
𝑝

𝑘 − 1
)

𝑝+1
𝑘=1 𝑦

𝑘 − 𝑦2 − 𝑦𝑝−2. 

 

Proof. Proof is obvious.     
ℎ𝑐 (𝐿𝑝,1, 𝑘) for 4 ≤  𝑝 ≤  14 and 1 ≤  𝑘 ≤  15. 

𝑘 
𝑝 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

4 2 4 7 4 1           

5 2 5 10 11 5 1          

6 2 6 15 20 16 6 1         

7 2 7 21 35 35 22 7 1        

8 2 8 28 56 70 56 29 8 1       

9 2 9 36 84 126 126 84 37 9 1      

10 2 10 45 120 210 252 210 120 46 10 1     

11 2 11 55 165 330 462 462 330 165 56 11 1    

12 2 12 66 220 495 792 924 792 495 220 67 12 1   

13 2 13 78 286 715 1287 1716 1716 1287 715 286 79 13 1  

14 2 14 91 364 100 2002 3003 3432 3003 2002 1001 364 92 14 1 

Table 1 

 

Theorem 3.6 The coefficients of  𝐻𝑐 (𝐿𝑝,1, 𝑦) satisfy the following properties.  

  (i) ℎ𝑐 (𝐿𝑝,1, 𝑝 + 1) = 1, for every 𝑝 ≥ 4. 

  (ii) ℎ𝑐 (𝐿𝑝,1, 𝑝) = 𝑝, for every 𝑝 ≥ 4. 

  (iii) ℎ𝑐 (𝐿𝑝,1, 𝑝 − 1) =
1

2
(𝑝2 − 𝑝 + 2), for every 𝑝 ≥ 4. 

  (iv) ℎ𝑐 (𝐿𝑝,1, 𝑝 − 2) =
1

6
(𝑝3 − 3𝑝2 + 2𝑝), for every 𝑝 ≥ 4. 

  (v) ℎ𝑐 (𝐿𝑝,1, 𝑝 − 3) =
1

24
(𝑝4 − 6𝑝3 + 11𝑝2 − 6𝑝), for every 𝑝 ≥ 6. 

  (vi) ℎ𝑐 (𝐿𝑝,1, 1) =  2, for every 𝑝 ≥ 4. 

  (vii) ℎ𝑐 (𝐿𝑝,1, 2) =  𝑝, for every 𝑝 ≥ 4. 

 

 

4. Conclusion 

   In this paper, we identified the connected hub sets of 𝐿𝑝,1   and using the 

connected hub sets we derived the connected hub polynomial of 𝐿𝑝,1.   We can 

generalize this study to derive the connected hub polynomial of any Lollipop 

graph 𝐿𝑝,𝑞 . 

 

  

34



 Connected Hub Sets and Connected Hub Polynomials of the Lollipop Graph 𝐿𝑝,1 

 

 
 

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