Ratio Mathematica                                                                                   Volume 45, 2023 

 

 

 

Monophonic Distance Energy for Join of 

Some Graphs 

 
Sinju Manohar V S1  

 Binu Selin T2 
 

 

Abstract 

Let 𝐺 be a connected graph with 𝑛 vertices and 𝑚 edges. Let µ1, µ2 ,..., µn be the Eigen 
values of distance matrix of 𝐺. The distance energy of a graph ED (G) = ∑ |𝜇𝑖

 |
𝑝
𝑖=1 ,was 

already studied. We now define and investigate the monophonic distance energy as EM 

(G) = ∑ |𝜇𝑖
𝑀|𝑛𝑖=1 , where µ1

M, µ2
M..., µn

M are the eigen values of monophonic distance 

matrix of graphs. In this paper we find the monophonic distance energy for join of some 

graphs. 

 

Keywords: Join of graphs; Monophonic distance matrix; Monophonic distance energy. 

 

2010 Mathematics Subject Classification: 05C12, 05B203. 

 

 

 

 

 

 

 

 

 

 

 

 

 
1 Research Scholar, Reg No: 20113162092016, Department of Mathematics, Scott Christian College 

(Autonomous), Nagercoil - 629 003, India. (Affiliated to Manonmaniam Sundaranar University, 

Abishekapatti, Tirunelveli-627 012, Tamil Nadu, India.). Email: sinjusinju2124@gmail.com 
2 Assistant Professor, Department of Mathematics, Scott Christian College (Autonomous), Nagercoil - 629 

003, India. (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627 012, Tamil 

Nadu, India.). Email: binuselin@gmail.com 
3 Received on July 19, 2022. Accepted on October 15, 2022. Published on January 30, 2023. doi: 

10.23755/rm.v45i0.979. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY license agreement. 

60



Sinju Manohar V S, Binu Selin T 

 

 

1. Introduction 

In this paper we considered simple, connected and undirected graphs. The concept of 

energy of a graph was introduced by I. Gutman [2] in the year 1978. Let 𝐺 be a connected 
graph with 𝑛 vertices and 𝑚 edges. Let 𝐴 = (𝑎𝑖𝑗 ) be the adjacency matrix of the graph. 

The eigenvalues λ1, λ2, ..., λn of A, assumed in non-increasing order, are the eigen values 

of the graph 𝐺.  The energy 𝐸(𝐺) of 𝐺 is defined to be the sum of the absolute values 
of the eigen values of 𝐺.  ie., 𝐸(𝐺) = ∑ |𝜆𝑖

 |𝑛𝑖=1 . Also, Distance energy of a graph was 
introduced by I. Gutman and others [4] in the year 2008.  A. P. Santhakumaran and 

others introduced the monophonic number of a graph in 2014 [8]. For any two vertices 𝑢 
and 𝑣 in a connected graph𝐺, a 𝑢 − 𝑣 path is a monophonic path if it contains no chords, 
and the monophonic distance 𝑑m(𝑢, 𝑣) is the length of a longest 𝑢 − 𝑣 monophonic path 
in 𝐺. Based on these we introduce a new concept monophonic distance energy of a graph. 
Based on these we introduce a new concept monophonic distance energy of a graph. In 

this paper we investigate the monophonic distance energy of 𝐾1,𝑛 + 𝐾1,𝑛, 𝐾𝑛,𝑛 + 𝐾𝑛, 
𝐾𝑛,𝑛 + 𝐾𝑛,𝑛 and 𝐾𝑛 + 𝐾1,𝑛. 

 

2. Definitions  

Definition 2.1. Let 𝐺 be a connected graph with vertex set 𝑣1, 𝑣2, . . . , 𝑣n. The monophonic 
distance matrix of 𝐺 is defined as  

𝑀 = 𝑀 [𝐺] =  (𝑑𝑚𝑖𝑗 ) 𝑛×𝑛   Where 𝑑𝑚𝑖𝑗 =  {
𝑑𝑚 (𝑣𝑖 , 𝑣𝑗 ),            𝑖𝑓 𝑖 ≠ j

0,                       𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
. 

Here 𝑑m(𝑣i, 𝑣j) is the monophonic distance of 𝑣i to 𝑣j . The eigen values of Monophonic 
distance matrix 𝑀(𝐺) are denoted by µ1

M, µ2
M ..., µn

M and are said to be 𝑀-Eigen values 
of 𝐺  and to form the 𝑀-spectrum of 𝐺, denoted by 𝑠𝑝𝑒𝑐M (𝐺). We note that since the 
Monophonic distance matrix is symmetric, its eigen values are real and can be ordered as 

µ1
M ≤ µ2

M ≤ ... ≤ µn
M. We can define the monophonic distance energy of a graph as 

𝐸M(𝐺)  =  ∑ |𝜇𝑖
𝑀|𝑛𝑖=1 . 

 

3. Main Results 

Definition 3.1. [3] The join 𝐺 = 𝐺1 + 𝐺2 of graphs 𝐺1 and 𝐺2 with disjoint point sets 𝑉1 
and 𝑉2 and edge sets 𝑋1 and 𝑋2 is the graph union 𝐺1 ∪ 𝐺2 together with all the edges 
joining  𝑉1 and 𝑉2. 
 

Theorem 3.2. For the Star graph 𝐾1,𝑛, 𝐸𝑀(𝐾1,𝑛 + 𝐾1,𝑛) = 5(𝑛 − 1) + √9𝑛
2 − 2𝑛 + 9. 

Proof.  From the definition of join, the monophonic distance matrix of 𝐾1,𝑛 + 𝐾1,𝑛 can be 
written as 

61



Monophonic Distance Energy for Join of Some Graphs 
 

𝑀(𝐾1,𝑛 + 𝐾1,𝑛)  =  [
𝑀(𝐾1,𝑛) 𝐽𝑛

𝐽𝑛 𝑀(𝐾1,𝑛)
]  

where 𝑀(𝐾1,𝑛) be a monophonic distance matrix of 𝐾1,𝑛. 
We have 𝑀(𝐾1,𝑛 + 𝐾1,𝑛) = (𝑑𝑚𝑖𝑗 )2(𝑛+1)×2(𝑛+1). 

The monophonic distance spectrum 𝑆𝑝𝑒𝑐M (𝐾1,𝑛 + 𝐾1,𝑛) is  

( −2 −1
(3𝑛 − 1) − √9𝑛2 − 2𝑛 + 9

2
(𝑛 − 1) 1 1

   (𝑛 − 2)
(3𝑛 − 1) + √9𝑛2 − 2𝑛 + 9

2
1 1

). 

Thus the monophonic distance energy of   𝐾1,𝑛 + 𝐾1,𝑛 is  

𝐸M(𝐾1,𝑛 + 𝐾1,𝑛) = ∑ |𝜇𝑖
𝑀|

2(𝑛+1)
𝑖=1

,where 𝜇1
𝑀, 𝜇2

𝑀 , … , 𝜇2(𝑛+1)
𝑀 are the eigen values of 

monophonic distance matrix 𝑀(𝐾1,𝑛 + 𝐾1,𝑛). 
For 𝑛 >  1, 

𝐸M(𝐾1,𝑛 + 𝐾1,𝑛) =  |−2| + |−2| + ⋯ + |−2| + |−1| 

+ |
(3𝑛 − 1) − √9𝑛2 − 2𝑛 + 9

2
| + |(𝑛 − 2)| 

+ |
(3𝑛 − 1) + √9𝑛2 − 2𝑛 + 9

2
| 

=  2 (2(𝑛 − 1) + 1 −
(3𝑛 − 1) + √9𝑛2 − 2𝑛 + 9

2
+ (𝑛 − 2)

+
(3𝑛 − 1) + √9𝑛2 − 2𝑛 + 9

2
 

=  5(𝑛 − 1) + √9𝑛2 − 2𝑛 + 9. 
 

Theorem 3.3. For the complete bipartite graph 𝐾𝑛,𝑛 and complete graph 𝐾𝑛,  

𝐸𝑀(𝐾𝑛,𝑛 + 𝐾𝑛) =  {
(6𝑛 − 7) + √12𝑛2 − 4𝑛 + 1 𝑖𝑓 1 ≤ 𝑛 ≤ 4

10(𝑛 − 1) 𝑖𝑓 𝑛 ≥ 5
. 

Proof.  From the definition of join, the monophonic distance matrix of 𝐾𝑛,𝑛 + 𝐾𝑛 can be 
written as 

𝑀(𝐾𝑛,𝑛 + 𝐾𝑛)  =  [
𝑀(𝐾𝑛,𝑛) 𝐽2𝑛,𝑛

𝐽𝑛,2𝑛 𝑀(𝐾𝑛)
]  

where 𝑀(𝐾𝑛,𝑛) be a monophonic distance matrix of 𝐾𝑛,𝑛 and M(𝐾𝑛) be a monophonic distance 
matrix of 𝐾𝑛. 
We have 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛) = (𝑑𝑚𝑖𝑗 )3𝑛×3𝑛. 

The monophonic distance spectrum 𝑆𝑝𝑒𝑐M (𝐾𝑛,𝑛 + 𝐾𝑛) is  

( −2 −1
(4𝑛 − 3) − √12𝑛2 − 4𝑛 + 1

2
2(𝑛 − 1) (𝑛 − 1) 1

   
(4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1

2
 
𝑛 − 2

1
1

). 

Thus the monophonic distance energy of   𝐾𝑛,𝑛 + 𝐾𝑛 is  

62



Sinju Manohar V S, Binu Selin T 

𝐸M(𝐾𝑛,𝑛 + 𝐾,𝑛) = ∑ |𝜇𝑖
𝑀|3𝑛𝑖=1 ,where 𝜇1

𝑀, 𝜇2
𝑀 , … , 𝜇3𝑛

𝑀 are the eigen values of 

monophonic distance matrix 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛). 
For 1 ≤ 𝑛 ≤ 4, 
𝐸M(𝐾𝑛,𝑛 + 𝐾𝑛) =  |−2| + |−2| + ⋯ + |−2| + |−1| + |−1| + ⋯ + |−1| 
 

+ |
(4𝑛 − 3) − √12𝑛2 − 4𝑛 + 1

2
| + |

(4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1

2
| + (𝑛 − 2) 

=  2 (2(𝑛 − 1) + (𝑛 − 1) −
(4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1

2

+
(4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1

2
+ (𝑛 − 2) 

=  (6𝑛 − 7) + √12𝑛2 − 4𝑛 + 1 
For 𝑛 ≥ 5, 
     𝐸M(𝐾𝑛,𝑛 + 𝐾𝑛) =  |−2| + |−2| + ⋯ + |−2| + |−1| + |−1| + ⋯ + |−1| 

 

+ |
(4𝑛 − 3) − √12𝑛2 − 4𝑛 + 1

2
| + |

(4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1

2
| + (𝑛 − 2) 

=  2 (2(𝑛 − 1) + (𝑛 − 1) +
(4𝑛 − 3) − √12𝑛2 − 4𝑛 + 1

2

+
(4𝑛 − 3) + √12𝑛2 − 4𝑛 + 1

2
+ (n − 2) 

=  10(𝑛 − 1). 
 

Theorem 3.4. For the complete bipartite graph 𝐾𝑛,𝑛 ,  

𝐸𝑀(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) =  {
2(5𝑛 − 2) 𝑖𝑓 𝑛 = 1
16(𝑛 − 1) 𝑖𝑓 𝑛 ≥ 2

. 

Proof.  From the definition of join, the monophonic distance matrix of 𝐾𝑛,𝑛 + 𝐾𝑛,𝑛 can 
be written as 

𝑀(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛)  =  [
𝑀(𝐾𝑛,𝑛) 𝐽2𝑛

𝐽2𝑛 𝑀(𝐾𝑛,𝑛)
]  

where 𝑀(𝐾𝑛,𝑛) be a monophonic distance matrix of 𝐾𝑛,𝑛  
We have 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) = (𝑑𝑚𝑖𝑗 )4𝑛×4𝑛. 

The monophonic distance spectrum 𝑆𝑝𝑒𝑐M(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) is  

(
−2 (𝑛 − 2) (5𝑛 − 2)

4(𝑛 − 1) 3 1
   ). 

Thus the monophonic distance energy of   𝐾𝑛,𝑛 + 𝐾𝑛,𝑛 is  

𝐸M(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) = ∑ |𝜇𝑖
𝑀|4𝑛𝑖=1 ,where 𝜇1

𝑀, 𝜇2
𝑀, … , 𝜇4𝑛

𝑀  are the eigen values of 

monophonic distance matrix 𝑀(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛). 
For 𝑛 = 1, 

63



Monophonic Distance Energy for Join of Some Graphs 
 

𝐸M(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) =  |−2| + |−2| + ⋯ + |−2| + |𝑛 − 2| + ⋯ + |𝑛 − 2| + |5𝑛 − 2| =

 2 (4(𝑛 − 1) + 3(−𝑛 + 2) + 5𝑛 − 2 =  8𝑛 − 8 − 3𝑛 + 6 + 5𝑛 − 2 
= 10𝑛 − 4 
= 2(5𝑛 − 2) 
For 𝑛 ≥ 2, 
𝐸M(𝐾𝑛,𝑛 + 𝐾𝑛,𝑛) =  |−2| + |−2| + ⋯ + |−2| + |𝑛 − 2| + ⋯ + |𝑛 − 2| + |5𝑛 − 2| =

 2 (4(𝑛 − 1) + 3(𝑛 − 2) + 5𝑛 − 2 =  8𝑛 − 8 + 3𝑛 − 6 + 5𝑛 − 2 
= 16𝑛 − 16 
= 16(𝑛 − 1). 
 

Theorem 3.5. For the star graph 𝐾1,𝑛 and complete graph 𝐾𝑛,  

𝐸𝑀(𝐾𝑛 + 𝐾1,𝑛) =  {
(3𝑛 − 2) + √5𝑛2 + 4 𝑖𝑓 𝑛 ≤ 2

2(3𝑛 − 2) 𝑖𝑓 𝑛 ≥ 3
. 

Proof. . From the definition of join, the monophonic distance matrix of 𝐾𝑛 + 𝐾1,𝑛 can be 
written as 

𝑀(𝐾𝑛 + 𝐾1,𝑛)  =  [
𝑀(𝐾𝑛) 𝐽𝑛,𝑛+1
𝐽𝑛+1,𝑛 𝑀(𝐾1,𝑛)

]  

where 𝑀(𝐾1,𝑛) be a monophonic distance matrix of 𝐾1,𝑛 and M(𝐾𝑛) be a monophonic distance 
matrix of 𝐾𝑛. 
We have 𝑀(𝐾𝑛 + 𝐾1,𝑛) = (𝑑𝑚𝑖𝑗 )(2𝑛+1)×(2𝑛+1). 

The monophonic distance spectrum 𝑆𝑝𝑒𝑐M (𝐾𝑛,𝑛 + 𝐾𝑛) is  

( −2 −1
(3𝑛 − 2) − √5𝑛2 + 4

2
(𝑛 − 1) 𝑛 1

   
(3𝑛 − 2) + √5𝑛2 + 4

2
1

). 

Thus the monophonic distance energy of   𝐾𝑛 + 𝐾1,𝑛 is  

𝐸M(𝐾𝑛 + 𝐾1,𝑛) = ∑ |𝜇𝑖
𝑀|2𝑛+1𝑖=1 ,where 𝜇1

𝑀, 𝜇2
𝑀 , … , 𝜇2𝑛+1

𝑀 are the eigen values of 

monophonic distance matrix 𝑀(𝐾𝑛 + 𝐾1,𝑛). 
For 𝑛 ≤ 2, 
𝐸M(𝐾𝑛 + 𝐾1,𝑛) =  |−2| + |−2| + ⋯ + |−2| + |−1| + |−1| + ⋯ + |−1| 
 

+ |
(3𝑛 − 2) − √5𝑛2 + 4

2
| + |

(3𝑛 − 2) + √5𝑛2 + 4

2
| 

=  2 (𝑛 − 1) + 𝑛 −
(3𝑛 − 2) + √5𝑛2 + 4

2
+

(3𝑛 − 2) + √5𝑛2 + 4

2
 

= 2𝑛 − 2 + 𝑛 + √5𝑛2 + 4 

= (3𝑛 − 2) + √5𝑛2 + 4. 
For 𝑛 ≥ 5, 
𝐸M(𝐾𝑛 + 𝐾1,𝑛) =  |−2| + |−2| + ⋯ + |−2| + |−1| + |−1| + ⋯ + |−1| 
 

64



Sinju Manohar V S, Binu Selin T 

+ |
(3𝑛 − 2) − √5𝑛2 + 4

2
| + |

(3𝑛 − 2) + √5𝑛2 + 4

2
| 

=  2 (𝑛 − 1) + 𝑛 +
(3𝑛 − 2) − √5𝑛2 + 4

2
+

(3𝑛 − 2) + √5𝑛2 + 4

2
 

= 2𝑛 − 2 + 𝑛 + 3𝑛 − 2 
= 6𝑛 − 4 
= 2(3𝑛 − 2). 

 

4. Conclusions 
In this paper we discussed the concept of monophonic distance energy and obtained some 

results on monophonic distance energy for certain Join graphs. In future we planned to 

extend our research for various graph operations. 

 

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[5] Indulal. G and Vijayakumar. A, On a pair of Equienergetic Graphs, MATCH 

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