Ratio Mathematica                                                                           Volume 44,2022 

 
 

 

   

 

 

Reach Energy of Digraphs 

 
  

V. Mahalakshmi*  

B. Vijaya Praba†   

K. Palani‡ 
 

 

Abstract 

A Digraph D consists of two finite sets (𝑉, 𝒜), where 𝑉 denotes the vertex set and 
𝒜 denotes the arc set. For vertices 𝑢, 𝑣 ∈ 𝑉, if there exists a directed path from 𝑢 to 𝑣 
then 𝑣 is said to be reachable from 𝑢 and vice versa. The Reachability matrix of D is the 
𝑛 × 𝑛 matrix 𝑅(𝐷) = [𝑟𝑖𝑗], where 𝑟𝑖𝑗 = 1, if 𝑣𝑗 is reachable from 𝑣𝑖 and 𝑟𝑖𝑗 = 0 

otherwise. The eigen values corresponding to the reachability matrix are called reach 

eigen values. The reach energy of a digraph is defined by 𝐸𝑅(𝐷) = ∑ |𝜆𝑖|
𝑛
𝑖=1 where 𝜆𝑖 is 

the eigen value of the reachability matrix. In this paper we introduce the reach spectrum 

of a digraph and study its properties and bounds. Moreover, we compute reach spectrum 

for some digraphs. 

 

Keywords: Reachable, Reachability matrix, reach eigen values, reach spectrum, Reach 

energy. 

 

2010 AMS subject classification: 05C50,05C90, 15A18§ 

 

 
*Assistant Professor (PG & Research Department of Mathematics, A.P.C. Mahalaxmi College for Women 

Thoothukudi, India); Email - mahalakshmi@apcmcollege.ac.in 
†Research Scholar Reg.No: 21212012092002 (PG & Research Department of Mathematics, A.P.C. 

Mahalaxmi College for Women Thoothukudi, India); Email - vijayapraba.b@gmail.com 
‡ Third Author’s Affiliation (PG & Research Department of Mathematics, A.P.C. Mahalaxmi College for 

Women Thoothukudi, India); Email - palani@apcmcollege.ac.in 

Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamilnadu, India; 
§Received on June 19th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 

10.23755/rm.v44i0.926. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY licence agreement. 

 

363



 

V. Mahalakshmi, B. Vijaya Praba, and K. Palani 

 

 

1. Introduction 

In this paper we considered simple and connected graph. A directed graph or 

digraph D consists of two finite sets (𝑉, 𝒜 ) where 𝑉 denotes the vertex set and 𝒜 
denotes the arc set. For two vertices 𝑢 and 𝑣, an arc from 𝑢 to 𝑣 is denoted by 𝑢𝑣. Two 
vertices 𝑢 and 𝑣 is said to be adjacent if either 𝑢𝑣 ∈  𝒜 or 𝑣𝑢 ∈ 𝒜 . In 1978 Gutman [4] 
defined the energy of a simple graph as the sum of the absolute values of its eigen 

values and it is denoted by 𝐸(𝐺). i.e., 𝐸(𝐺) = ∑ |𝜆𝑖|
𝑛
𝑖=1 . The concept of graph energy 

was extended to digraph by Pena and Rada [8] and Adiga et al. [1]. Khan et al. [5] 

defined a new notion of energy of digraph called iota energy. In this paper, we 

investigate the properties and some bounds on reach energy. 

 

Definition 1.1. A path is said to be directed path in which all the edges are directed 

either in clockwise or in anticlockwise direction and it is denoted as 𝑃𝑛⃗⃗⃗⃗  ⃗. Let 
{𝑣1, 𝑣2, … , 𝑣𝑛} be the vertex set of a directed path. Then the set {𝑣𝑖𝑣𝑖+1|  𝑖 = 1,2, . . . , 𝑛 −

1} is the arc set of 𝑃𝑛⃗⃗⃗⃗  ⃗.  
 

Definition 1.2. A path is said to be alternate path in which the edges are given alternate 

direction and it is denoted as 𝐴𝑃𝑛⃗⃗ ⃗⃗⃗⃗ ⃗⃗  
 

Definition 1.3. A star graph 𝐾1,𝑛 in which all the edges are directed towards the root 

vertex is called an instar and is denoted as 𝑖𝐾1,𝑛⃗⃗⃗⃗⃗⃗⃗⃗  ⃗ 
 

Definition 1.4. A star graph 𝐾1,𝑛 in which all the edges are directed away from the root 

vertex is called an outstar and is denoted as 𝑜𝐾1,𝑛⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗  
 

2. Reach Energy 

Definition 2.1. Let 𝐷 = (𝑉, 𝐴) be a directed graph with 𝑛 vertices. The reachability 
matrix [2] 𝑅(𝐷) = [𝑟𝑖𝑗] is the 𝑛 × 𝑛 matrix with 𝑟𝑖𝑗 = 1, if 𝑣𝑗  is reachable from 𝑣𝑖 and 

𝑟𝑖𝑗 = 0 otherwise. We assume that each vertex is reachable from itself. The 

characteristic polynomial of 𝑅(𝐷) = [𝑟𝑖𝑗] is denoted by 𝑓(𝐷, 𝜆) = 𝑑𝑒𝑡(𝑅(𝐷) − 𝐼𝜆). Let 

{𝜆1, 𝜆2, … , 𝜆𝑛} be the reach eigen values of 𝐷. The reach eigen values of the graph 𝐷 are 
the eigen values of 𝑅(𝐷) and is called as reach spectrum of 𝐷. The spectrum of D is 
denoted by  

𝑠𝑝𝑒𝑐 𝐷 = {
𝜆1  𝜆2  … 𝜆𝑛
𝑚1 𝑚2 … 𝑚𝑛

} 

where 𝑚𝑖 is the algebraic multiplicity of the eigen values 𝜆𝑖, for 1 ≤ 𝑖 ≤ 𝑛 
Then the reach energy of 𝐷 is defined as the sum of absolute values of reach spectrum 
of 𝐷. 
i.e., 𝐸𝑅(𝐷) = ∑ |𝜆𝑖|

𝑛
𝑖=1  

 

364



 

Reach energy of digraphs 

 

 

 

Example 2.2. 

 

 

 

 

 

 

 

 
Figure 1. 

 

Reachability matrix is 

 

𝑅(𝐷) =

(

  
 

1 1 1 1 1 1

0 1 1 1 1 1

0 0 1 1 0 1

0 0 0 1 0 0

0 0 0 1 1 0

0 0 0 0 0 1)

  
 

 

 

Characteristic polynomial of 𝑅(𝐷) is given by 
𝑓(𝐷, 𝜆) = 𝜆6 − 6𝜆5 + 15𝜆4 − 20𝜆3 + 15𝜆2 − 6𝜆 + 1. 

Hence, the reach spectrum is {
1
6
}. 

Therefore, the reach energy of D is 𝐸𝑅(𝐷) = 6. 
 

3. Reach Energy of Some Graphs 

Theorem 3.1. Directed path and alternate path attains same Reach Energy. 

Proof: Let 𝑃𝑛(𝐷) be the directed path with vertex set 𝑉 = {𝑣1, 𝑣2, … , 𝑣𝑛}  
Let 𝐴𝑃𝑛(𝐷) be the alternate path with vertex set 𝑉 = {𝑣1′, 𝑣2′, … , 𝑣𝑛′}. 
The reachability matrix of 𝑃𝑛(𝐷) is in the upper triangular matrix form with the entries 
1. 

The reachability matrix of 𝐴𝑃𝑛(𝐷) is of the form 

𝑅(𝐴𝑃𝑛(𝐷)) =  

[
 
 
 
 
 
1 1 0 ⋯ 0 0
0 1 0 ⋯ 0 0
0 1 ⋯ 1 0 0
0 0 0 ⋱ 1 0
⋮ ⋮ ⋮ 0 ⋮ ⋮
0 0 ⋯ 0 1 1]

 
 
 
 
 

 

𝑣1 𝑣2 𝑣3 𝑣4 

𝑣5 
𝑣6 

365



 

V. Mahalakshmi, B. Vijaya Praba, and K. Palani 

 

 

 

Characteristic polynomial of 𝑃𝑛 is 𝑓(𝑃𝑛(𝐷), 𝜆) = det(𝑅(𝑃𝑛(𝐷)) − 𝜆 𝐼𝑛) 
 

𝑓(𝑃𝑛(𝐷), 𝜆) = |

1 − 𝜆 1 ⋯ 1
0 1 − 𝜆 1 ⋮
⋮ ⋮ ⋱ 1
0 ⋯ 0 1 − 𝜆

| 

 

= (−1)𝑛 (𝜆 − 1)𝑛. 
 

Characteristic polynomial of 𝐴𝑃𝑛 is 𝑓(𝐴𝑃𝑛(𝐷), 𝜆) = det(𝑅(𝐴𝑃𝑛(𝐷)) − 𝜆 𝐼𝑛) 
 

𝑓(𝐴𝑃𝑛(𝐷), 𝜆) =
|

|

1 − 𝜆 1 0 ⋯ 0 0
0 1 − 𝜆 0 ⋯ 0 0
0 1 ⋯ 1 0 0
0 0 0 ⋱ 1 0
⋮ ⋮ ⋮ 0 ⋮ ⋮
0 0 ⋯ 0 1 1 − 𝜆

|

|
 

 

= (−1)𝑛 (𝜆 − 1)𝑛. 
Clearly, 𝑓(𝑃𝑛(𝐷), 𝜆) = 𝑓(𝐴𝑃𝑛(𝐷), 𝜆). 
 

Since the characteristic polynomial of 𝑃𝑛(𝐷) and 𝐴𝑃𝑛(𝐷) are same, Spectrum of 𝑅(𝑃𝑛) 
and 𝑅(𝐴𝑃𝑛) are same. 

Hence, the Reach Spectrum of 𝑅(𝑃𝑛(𝐷)) and 𝑅(𝐴𝑃𝑛(𝐷)) are {
1
𝑛
} and its Reach Energy 

is 

𝐸𝑅(𝑃𝑛(𝐷)) = 𝐸𝑅(𝐴𝑃𝑛(𝐷)) = ∑ 1

𝑛

1

= 𝑛 

Therefore, the directed path and alternate path attains same Reach Energy. 

 

Theorem 3.2. Reach energy of directed star is independent of its orientation. 

Proof: Let 𝐾1,𝑛−1(𝐷) be the directed instar with vertex set 𝑣1, 𝑣2, … , 𝑣𝑛−1  
Let 𝐾′1,𝑛−1(D) be the directed outstar with vertex set 𝑣1′, 𝑣2′, … , 𝑣𝑛−1′  
The reachability matrix of 𝐾1,𝑛−1(𝐷) is of the form 

𝑅 (𝐾1,𝑛−1(𝐷)) = 𝐼𝑛 +  (
0 01×𝑛−1

𝐽𝑛−1×1 0𝑛−1×𝑛−1
) 

 

The reachability matrix of 𝐾′1,𝑛−1(𝐷) is of the form 

𝑅 (𝐾′1,𝑛−1(𝐷)) = 𝐼𝑛 +  (
0 𝐽1×𝑛−1

0𝑛−1×1 0𝑛−1×𝑛−1
) 

Characteristic polynomial of 𝐾1,𝑛−1(𝐷) is  

366



 

Reach energy of digraphs 

 

 

𝑓(𝐾1,𝑛−1(𝐷), 𝜆) = det(𝑅 (𝐾1,𝑛−1(𝐷)) − 𝜆 𝐼𝑛) 

𝑓(𝐾1,𝑛−1(𝐷), 𝜆) = |

1 − 𝜆 0 ⋯ 0
1 1 − 𝜆 0 ⋮
⋮ 0 ⋱ 0
1 0 0 1 − 𝜆

| 

 

= (−1)𝑛 (𝜆 − 1)𝑛 
 

Characteristic polynomial of 𝐾′1,𝑛−1(𝐷) is  
 

𝑓(𝐾′1,𝑛−1(𝐷), 𝜆) = det(𝑅 (𝐾′1,𝑛−1(𝐷)) − 𝜆 𝐼𝑛) 

 

𝑓(𝐾′1,𝑛−1(𝐷), 𝜆) = |

1 − 𝜆 1 ⋯ 1
0 1 − 𝜆 0 0
⋮ 0 ⋱ 0
0 ⋯ 0 1 − 𝜆

| 

 

= (−1)𝑛 (𝜆 − 1)𝑛. 
Clearly, 𝑓(𝐾1,𝑛−1(𝐷) , 𝜆) = 𝑓(𝐾′1,𝑛−1(𝐷) , 𝜆). 
 

Since the characteristic polynomial of 𝐾1,𝑛−1(𝐷)  and 𝐾′1,𝑛−1(𝐷) are same, Spectrum of 
𝑅(𝐾1,𝑛−1(𝐷) ) and 𝑅(𝐾′1,𝑛−1(𝐷) ) are same. 

Hence, the Reach Spectrum of 𝑅(𝐾1,𝑛−1(𝐷) ) and 𝑅(𝐾′1,𝑛−1(𝐷) ) are {
1
𝑛

 and its Reach 

Energy 

𝐸𝑅 (𝐾1,𝑛−1(𝐷)) = 𝐸𝑅 (𝐾′1,𝑛−1(𝐷)) = ∑ 1

𝑛

1

= 𝑛 

Therefore, the Reach Energy of directed star is independent of its orientation. 

 

 

4. Properties of Reach Eigen values 

Theorem 4.1: Let 𝐷 be any digraph. If 𝜆1, 𝜆2, … , 𝜆𝑛 are the reach eigen values of 𝑅(𝐷), 
then the following condition holds. 

i. ∑ 𝜆𝑖 = 𝑛
𝑛
𝑖=1  

ii. ∑ 𝜆𝑖
2𝑛

𝑖=1 = 𝑛 + 𝛼 + 𝛽;  
where 𝛼 = ∑ 𝑟𝑖𝑗𝑟𝑗𝑖𝑖>𝑗  and 𝛽 = ∑ 𝑟𝑖𝑗𝑟𝑗𝑖𝑖<𝑗  

Proof: 

i. Sum of eigen values of 𝑅(𝐷) is same as the trace of 𝑅(𝐷). 
i.e., ∑ 𝜆𝑖 = ∑ 𝑟𝑖𝑖

𝑛
𝑖=1

𝑛
𝑖=1  ; 

Since each vertex is reachable from itself, all the diagonal entries must be 1. 

= 1 + 1 + 1 + ⋯ + 1 (𝑛 times)  

367



 

V. Mahalakshmi, B. Vijaya Praba, and K. Palani 

 

 

Therefore, 

∑ 𝜆𝑖 = 𝑛

𝑛

𝑖=1

 

ii. Since, the sum of squares of the eigen values of R is the trace of [𝑅(𝐷)]2 

∑ 𝜆𝑖
2

𝑛

𝑖=1

= ∑ ∑ 𝑟𝑖𝑗 𝑟𝑗𝑖

𝑛

𝑖=1

𝑛

𝑖=1

 

= ∑ 𝑟𝑖𝑖𝑟𝑖𝑖

𝑛

𝑖=𝑗=1

+ ∑ 𝑟𝑖𝑗𝑟𝑗𝑖

𝑛

𝑖≠𝑗=1

 

= ∑(𝑟𝑖𝑖)
2

𝑖=𝑗

+ ∑ 𝑟𝑖𝑗𝑟𝑗𝑖
𝑖>𝑗

+ ∑ 𝑟𝑖𝑗𝑟𝑗𝑖
𝑖<𝑗

 

= 𝑛 + 𝛼 + 𝛽 ; where 𝛼 = ∑ 𝑟𝑖𝑗𝑟𝑗𝑖𝑖>𝑗  and 𝛽 = ∑ 𝑟𝑖𝑗𝑟𝑗𝑖𝑖<𝑗  
Therefore,  

∑ 𝜆𝑖
2

𝑛

𝑖=1

= 𝑛 + 𝛼 + 𝛽 

 

 

5. Bounds for Reach Energy  

Theorem 5.1: Let 𝐷 be a directed graph. Let Z be the absolute value of determinant of 
the reachability matrix 𝑅 of 𝐷 i.e., 𝑍 = |det 𝑅(𝐷)| 
Then  

𝑛√𝑛 + 𝛼 + 𝛽 ≤ 𝐸𝑅(𝐷) ≤ √(𝑛 + 𝛼 + 𝛽) + 𝑛(𝑛 − 1)𝑍
2/𝑛 

Proof:  

We know that Cauchy Schwarz inequality is  

(∑ 𝑎𝑖𝑏𝑖

𝑛

𝑖=1

)

2

≤ (∑ 𝑎𝑖

𝑛

𝑖=1

)

2

(∑ 𝑏𝑖

𝑛

𝑖=1

)

2

 

Put 𝑎𝑖 = 1, 𝑏𝑖 = |𝜆𝑖|  

(∑|𝜆𝑖|

𝑛

𝑖=1

)

2

≤ (∑ 1

𝑛

𝑖=1

)

2

(∑|𝜆𝑖|
2

𝑛

𝑖=1

) 

 

[𝐸𝑅(𝐷)]
2 ≤ 𝑛2(𝑛 + 𝛼 + 𝛽) 

 

𝐸𝑅(𝐷) ≤ 𝑛√𝑛 + 𝛼 + 𝛽                                                         (1) 
Since arithmetic mean is not smaller than geometric mean, we have 

368



 

Reach energy of digraphs 

 

 

1

𝑛(𝑛 − 1)
∑|𝜆𝑖||𝜆𝑗|

𝑖≠𝑗

≥ (∏|𝜆𝑖||𝜆𝑗|

𝑖≠𝑗

)

1

𝑛(𝑛−1)

 

= (∏|𝜆𝑖|
2(𝑛−1)

𝑛

𝑖=1

)

1

𝑛(𝑛−1)

 

= ∏|𝜆𝑖|
2

𝑛

𝑛

𝑖=1

 

= |∏ 𝜆𝑖

𝑛

𝑖=1

|

2

𝑛

 

= |det 𝑅(𝐷)|
2

𝑛  = 𝑍
2

𝑛 

Therefore, 

∑ |𝜆𝑖||𝜆𝑗|𝑖≠𝑗 ≥ 𝑛(𝑛 − 1) 𝑍
2

𝑛                                                                                                   (2) 

Now consider, 

[𝐸𝑅(𝐷)]
2 = (∑|𝜆𝑖|

𝑛

𝑖=1

)

2

 

= ∑|𝜆𝑖|
2

𝑛

𝑖=1

+ ∑|𝜆𝑖||𝜆𝑖|

𝑖≠𝑗

 

≥ (𝑛 + 𝛼 + 𝛽) + 𝑛(𝑛 − 1) 𝑍
2

𝑛);  𝑏𝑦 (2) 

Hence, 𝐸𝑅(𝐷) ≥
√(𝑛 + 𝛼 + 𝛽 ) + 𝑛(𝑛 − 1) 𝑍

2

𝑛 

From (1) and (2), 

𝑛√𝑛 + 𝛼 + 𝛽 ≤ 𝐸𝑅(𝐷) ≤ √(𝑛 + 𝛼 + 𝛽) + 𝑛(𝑛 − 1)𝑍
2/𝑛 

 

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V. Mahalakshmi, B. Vijaya Praba, and K. Palani 

 

 

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