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A General Characteristic Equation for Eigen Values and Energy of 

Cycle Graph 
 

Aadarsh Chaudhary1 and Kamesh Kumar2 
1Research Scholar, Department of Mathematics, Faculty of Engineering, Teerthanker Mahaveer 

University, Moradabad 

2Assistant Professor, Department of Mathematics, Faculty of Engineering, Teerthanker Mahaveer 

University, Moradabad 

Email: 1aadarshchaudhary277@gmail.com, 2drkamesh.engineering@tmu.ac.in  

 

 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

 

Abstract: The characteristic equation of a graph frequently appears in 

mathematical sciences, chemistry and physics. As to graph theorists, the 

characteristic equation tells information about the structural properties of a 

graph. In this paper, we find out the characteristic equations & eigen value 

of cycle Cn of n  vertices. 

Keywords: Characteristic Equation, Eigen value, Cycle graph, Structural 

properties. 

 

 

1. Introduction 

All considered graph in this paper are simple, undirected, and connected. Assume that G is a simple, 

finite, undirected graph with vertex set 𝑉(𝐺) and edge set 𝐸(𝐺). G has n vertices, and each 

vertex is identified by the label v1, v2, … , vn . The adjacency matrix X(G) of the graph G is a 

square matrix of order n, whose (i, j) entry is equal to 1 if the vertices vi and vj are adjacent 

and is equal to zero otherwise. Adjacent matrix will be symmetric binary matrix for simple 

graph and all entries along principal diagonal of X(G) are all 0’s for simple graph. (Brooks) 

The characteristic polynomial of the graph G along the adjacency matrix X(G) is det⁡(λIn −

X(G)), where In is the unit matrix of order n and will be denoted by Δ(G, λ). 

det⁡(λIn − X(G)) = 0 is the characteristic equation of graph G and the eigenvalues of a graph 

G are defined as the eigenvalues of its adjacency matrix X(G). ( Kumar et. al) So they are just 

the roots of the characteristic equation det ⁡(λIn − X(G)) = 0 i.e. Δ(G, λ) = 0. Since X(G) is a 

real symmetric matrix, so its eigenvalues are all real with sum equal to zero. Denoting them by 

λ1, λ2, … , λn  and as a whole, they are called the spectrum of G. For specifics, spectral aspects 

of graphs, including characteristics of the characteristic polynomial, have been thoroughly 

investigated, see (Brouwer & Haemers) 

Characteristic Polynomials  

Δ(𝐺, 𝜆) = 𝜆𝑛 − 𝑆1,𝑛𝜆
𝑛−1 + 𝑆2,𝑛𝜆

𝑛−2 − 𝑆3,𝑛𝜆
𝑛−3+. . . +(−1)𝑘𝑆𝑘,𝑛𝜆

𝑛−𝑘+. . . +(−1)𝑛𝑆𝑛,𝑛 

mailto:aadarshchaudhary277@gmail.com
mailto:drkamesh.engineering@tmu.ac.in


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Where 𝑆𝑘,𝑛 is the sum of the principal minors of order k in a matrix of order n.  

Since cycle graphs have so many uses in domains like computer science, biology, and 

chemistry, graph theory has explored cycle graphs extensively. Even with its significance, 

figuring out cycle graph eigenvalues and energy has proven to be a difficult undertaking. In 

this study, we provide an equation for the general characteristics that can be applied to any 

cycle graph to find its energy and eigenvalues. The goal of this research is to present a thorough 

explanation of the equation's development and its numerous uses. This work aims to address 

the research challenge of not having a generic equation for calculating the energy and 

eigenvalues of cycle graphs. We can improve our knowledge of cycle graphs and their 

applications by proving this equation. The steps taken to construct the characteristic equation, 

its derivation, and its applications in diverse sectors will be the key points of contention in this 

work. In summary, this study will advance the subject of graph theory by offering a general 

formula for calculating cycle graph energy and eigenvalues, which can have important 

applications across a range of domains. A graph's energy can be expressed as the total of the 

absolute values of its adjacency matrix's eigenvalues. Stated otherwise, the total of the 

eigenvalues' magnitudes represents it. In mathematical terms, the energy E(G) of the graph G 

is given by: if the eigenvalues of the adjacency matrix A are λ₁, λ₂, . . . , λₙ.  Then graph energy 

𝐸(𝐺) =∣ 𝜆1 ∣ +∣ 𝜆2 ∣ +. . . +∣ 𝜆𝑛 ∣ (Estrada & Benzi) 

When examining the properties of cycle graphs, many matrices are crucial. Among these are 

the seidel adjacency matrix Laplacian matrix, singles Laplacian matrix, and normalized 

Laplacian matrix. The invention of a general characteristic equation that makes it possible to 

calculate the energy and eigenvalues of cycle graphs is particularly intriguing. This emphasizes 

the generic formulas for a cycle graph's spectra, which are based on the cycle's length and 

vertex count (Stin et. al). A crucial source of information about a graph's structure and 

connectivity is its energy and eigenvalues. Researchers can gain a better understanding of the 

connections between these attributes and the general dynamics of the graph by creating a 

general characteristic equation for cycle graphs. (Rowlinson) addresses spectral properties of 

non-bipartite graphs with three distinct eigenvalues, focusing on conditions for G to be the cone 

over a strongly regular graph and analyzing scenarios involving one non-main eigenvalue and 

certain degree conditions. With precisely three different eigenvalues and a maximum of one 

for the second greatest eigenvalue, [(Cheng et. al), (Qi et al)] categorize the connected graphs. 

By shedding light on the structure and features of connected regular graphs with four different 

eigenvalues, (Huang & Huang) advanced the knowledge of spectral qualities and validates 

some discoveries regarding their non-existence (Li et al.). 

2. Characteristics of Cyclic Graph: 

Let Cn be a cycle of n vertices v1, v2, ..., vn and adjacent matrix of Cn denoted by X(Cn) defined 

as: 

X(Cn) ⁡=

[
 
 
 
 
 
 
0 1 0
1 0 1
0 1 0

⋯
0 0 1
0 0 0
0 0 0

⋮ ⋱ ⋮
0 0 0
0 0 0
1 0 0

⋯
0 1 0
1 0 1
0 1 0]

 
 
 
 
 
 

⁡⁡⁡𝑛×𝑛

 



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Its Characteristic Polynomials is ∆(𝐶𝑛, 𝜆) = det⁡(𝜆𝐼𝑛 − 𝑋(𝐶𝑛)) 

∆(𝐶𝑛, 𝜆) =

|

|

⁡⁡𝜆 −1 ⁡⁡0
⁡⁡−1 ⁡⁡𝜆 −1
⁡⁡0 −1 ⁡⁡𝜆

⋯
0 ⁡0 ⁡⁡−1
0 ⁡0 ⁡⁡⁡⁡⁡0
0 ⁡0 ⁡⁡⁡⁡⁡0

⋮ ⋱ ⋮
⁡⁡0 ⁡0 ⁡⁡⁡⁡⁡⁡0
⁡⁡0 ⁡0 ⁡⁡⁡⁡⁡0
−1 ⁡0 ⁡⁡⁡⁡⁡⁡0

⋯
⁡𝜆 −1 ⁡⁡0
−1 ⁡⁡𝜆 −1
⁡⁡0 −1 ⁡⁡𝜆

|

|

 

∆(𝐶𝑛, 𝜆) = 𝜆𝑛 − 𝑆1,𝑛𝜆
𝑛−1 + 𝑆2,𝑛𝜆

𝑛−2 − 𝑆3,𝑛𝜆
𝑛−3 +⋯+ (−1)𝑘𝑆𝑘,𝑛𝜆

𝑛−𝑘 +⋯+ (−1)𝑛𝑆𝑛,𝑛 

And Characteristic equation is ∆(𝐶𝑛, 𝜆) = det⁡(𝜆𝐼𝑛 − 𝑋(𝐶𝑛)) = 0⁡⁡ i.e. 

𝜆𝑛 − 𝑆1,𝑛𝜆
𝑛−1 + 𝑆2,𝑛𝜆

𝑛−2 − 𝑆3,𝑛𝜆
𝑛−3 +⋯+ (−1)𝑘𝑆𝑘,𝑛𝜆

𝑛−𝑘 +⋯+ (−1)𝑛𝑆𝑛,𝑛 = 0 

Where 𝑆𝑘,𝑛 is the sum of the principal minors of order k of cycle 𝐶𝑛 in an adjacent matrix X. 

Now we will try to find the values of 𝑆𝑘,𝑛. 

There are four cases to find 𝑆𝑘,𝑛. 

Case I: If  n ≡ 0(mod⁡4), then 

Sk,n = {
0,⁡⁡⁡k⁡is⁡odd⁡or⁡k = n,
−n⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡k = 2,

  

and 

𝑆𝑘,𝑛 = (−1)𝑘 2⁄ [(𝑘 + 1) + (−1)(𝑘−2) 2⁄ ⁡ ∑ 𝑆(𝑘 − 2, 𝑘 + 𝑚)

𝑛−2−𝑘

𝑚=0

] 

where k is even, 𝑘 ≠ 2 and 𝑘 < 𝑛 − 1. 

For example: Let 𝐶4 be a cycle of 4 vertices, adjacent matrix of 𝐶4 is. 

X(𝐶4) ⁡= [

0 1
1 0

0 1
1 0

0 1
1 0

0 1
1 0

]

⁡⁡⁡4×4

 

Characteristic equation is ∆(𝐶4, 𝜆) = det(𝜆𝐼4 − 𝑋(𝐶4)) = 0 

⇒ 𝜆4 − 𝑆1,4𝜆
3 + 𝑆2,4𝜆

2 − 𝑆3,4𝜆 + 𝑆4,4 = 0 

⇒ 𝜆4 − 4𝜆2 = 0 as 𝑆1,4 = 0, 𝑆2,4 = −4, 𝑆3,4 = 0⁡𝑎𝑛𝑑⁡𝑆4,4 = 0 

⇒ 𝜆 = 0, 0, −2⁡𝑎𝑛𝑑⁡2 

Case II: If,  ⁡n ≡ 1(mod⁡4), then 



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𝑆𝑘,𝑛 =

{
 

 
0,⁡⁡⁡⁡⁡⁡⁡k⁡is⁡odd⁡and⁡k ≠ n

−n,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡k = 2
2,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡k = n⁡

n,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡k = n − 1

 

And 

Sk,n=(−1)
𝑘 2⁄ [(𝑘 + 1) + (−1)(𝑘−2) 2⁄ ⁡ ∑ 𝑆(𝑘 − 2, 𝑘 + 𝑚)

𝑛−2−𝑘

𝑚=0

] 

 where k is even, 𝑘 ≠ 2 and 𝑘 < 𝑛 − 1. 

For example: Let 𝐶5 be a cycle of 5 vertices, adjacent matrix of 𝐶5 is. 

𝑋(𝐶5) = 

[
 
 
 
 
0 1 0 0 1
1 0 1 0 0
0
0
1

1
0
0

0
1
0

1
0
1

0
1
0]
 
 
 
 

⁡⁡⁡5×5

 

Characteristic equation is ∆(𝐶5, 𝜆) = det(𝜆𝐼5 − 𝑋(𝐶5)) = 0 

⇒ 𝜆5 − 𝑆1,5𝜆
4 + 𝑆2,5𝜆

3 − 𝑆3,5𝜆
2 + 𝑆4,5𝜆 − 𝑆5,5 = 0 

⇒ 𝜆5 − 5𝜆3 + 5𝜆 − 2 = 0 as 𝑆1,5 = 0, 𝑆2,5 = −5, 𝑆3,5 = 0, 𝑆4,5 = 5⁡𝑎𝑛𝑑⁡𝑆5,5 = 2 

⇒ 𝜆 = −1.6180,−1.6180, 0.6180, 0.6180, 2 

Case III: If 2(mod 4)n   and 𝑛 ≠ 2, then 

𝑆𝑘,𝑛 = {
0, 𝑘⁡𝑖𝑠⁡𝑜𝑑𝑑⁡𝑎𝑛𝑑⁡𝑘 ≠ 𝑛
−𝑛,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑘 = 2
−4,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑘 = 𝑛

 

And 

𝑆𝑘,𝑛 = (−1)𝑘 2⁄ [(𝑘 + 1) + (−1)(𝑘−2) 2⁄ ⁡ ∑ 𝑆(𝑘 − 2, 𝑘 + 𝑚)

𝑛−2−𝑘

𝑚=0

] 

where k is even, 𝑘 ≠ 2 and 𝑘 < 𝑛 − 1. 

For example: Let 𝐶6 be a cycle of 6 vertices, adjacent matrix of 𝐶6 is 

𝑋(𝐶6) ⁡=

[
 
 
 
 
 
0 1 0 0 0 1
1 0 1 0 0 0
0
0
0
1

1
0
0
0

0
1
0
0

1
0
1
0

0
1
0
1

0
0
1
0]
 
 
 
 
 

⁡⁡⁡6×6

 

Characteristic equation is ∆(𝐶6, 𝜆) = det(𝜆𝐼6 − 𝑋(𝐶6)) = 0 



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⇒ λ6 − S1,6λ
5 + S2,6λ

4 − S3,6λ
3 + S4,6λ

2 − S5,6λ + S6,6 = 0 

⇒ λ6 − 6λ4 + 9λ2 − 4 = 0 as S1,6 = 0, S2,6 = −6, S3,6 = 0, S4,6 = 9, S5,6 = 0⁡and⁡S6,6 = −4 

⇒ λ = −2,−1,−1, 1, 1, 2 

Case IV: If n ≡ 3(mod⁡4),⁡then 

𝑆𝑘,𝑛 =

{
 

 
0, 𝑘⁡𝑖𝑠⁡𝑜𝑑𝑑⁡𝑎𝑛𝑑⁡𝑘 ≠ 𝑛

−𝑛,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑘 = 2
2,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑘 = 𝑛

−𝑛,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑘 = 𝑛 − 1

 

And 

𝑆𝑘,𝑛 = (−1)𝑘 2⁄ [(𝑘 + 1) + (−1)(𝑘−2) 2⁄ ⁡ ∑ 𝑆(𝑘 − 2, 𝑘 + 𝑚)

𝑛−2−𝑘

𝑚=0

] 

where k is even, 𝑘 ≠ 2 and 𝑘 < 𝑛 − 1. 

For example: Let 𝐶7 be a cycle of 7 vertices, adjacent matrix of C7 is 

X(C7) ⁡=

[
 
 
 
 
 
 
0 1 0 0 0 0 1
1 0 1 0 0 0 0
0
0
0
0
1

1
0
0
0
0

0
1
0
0
0

1
0
1
0
0

0
1
0
1
0

0
0
1
0
1

0
0
0
1
0]
 
 
 
 
 
 

⁡⁡⁡7×7

 

Characteristic equation is ∆(𝐶7, 𝜆) = det(𝜆𝐼7 − 𝑋(𝐶7)) = 0 

⇒ 𝜆7 − 𝑆1,7𝜆
6 + 𝑆2,7𝜆

5 − 𝑆3,7𝜆
4 + 𝑆4,7𝜆

3 − 𝑆5,7𝜆
2 + 𝑆6,7𝜆 − 𝑆7,7 = 0 

⇒ 𝜆7 − 7𝜆5 + 14𝜆3 − 7𝜆 − 2 = 0 as 𝑆1,7 = 0, 𝑆2,7 = −7, 𝑆3,7 = 0, 𝑆4,7 = 14, 𝑆5,7 = 0, 𝑆6,7 =

−7⁡𝑎𝑛𝑑⁡𝑆7,7 = 2 

⇒ 𝜆 = −1.8019,−1.8019, −0.4450,−0.4450, 1.2470, 1.2470, 2 

Table: For 𝑺𝒌,𝒏⁡of cycle 𝑪𝒏 

n  𝑆1,𝑛
 
𝑆2,𝑛 𝑆3,𝑛

 
𝑆4,𝑛 𝑆5,𝑛 

𝑆6,𝑛 𝑆7,𝑛 
𝑆8,𝑛 𝑆9,𝑛 

𝑆10,𝑛 𝑆12,𝑛 𝑆14,𝑛 𝑆16,𝑛 𝑆18,𝑛 𝑆20,𝑛 

3 0 -3 2             

4 0 -4 0 0            

5 0 -5 0 5 2           

6 0 -6 0 9 0 -4          

7 0 -7 0 14 0 -7 2         

8 0 -8 0 20 0 -16 0 0        

9 0 -9 0 27 0 -30 0 9 2       

10 0 -10 0 35 0 -50 0 25 0 -4      



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11 0 -11 0 44 0 -77 0 55 0 -11      

12 0 -12 0 54 0 -112 0 105 0 -36 0     

13 0 -13 0 65 0 -156 0 182 0 -91 13     

14 0 -14 0 77 0 -210 0 294 0 -196 49 -4    

15 0 -15 0 90 0 -275 0 450 0 -378 140 -15    

16 0 -16 0 104 0 -352 0 660 0 -672 336 -64 0   

17 0 -17 0 119 0 -442 0 935 0 -1122 714 -204 17   

18 0 -18 0 135 0 -546 0 1287 0 -1782 1386 -540 81 -4  

19 0 -19 0 152 0 -665 0 1729 0 -2717 2508 -1254 285 -19  

20 0 -20 0 170 0 -800 0 2275 0 -4004 4290 -2640 825 -100 0 

 

Table: For Eigen values and Energy of cycle nC  

n  Eigen values Energy 

3 -1, 1, 2 4 

4 -2, 0, 0, 2 4 

5 -1.6180, -1.6180, 0.6180, 0.6180, 2 6.472 

6 -2, -1, -1, 1, 1, 2 8 

7 -1.8019, -1.8019, -0.4450, -0.4450, 1.2470, 1.2470, 2 8.9878 

8 -2, -1.4142, -1.4142, 0, 0, 1.4142, 1.4142, 2 9.6568 

9 -1.8794, -1.8794, -1, -1, 0.3473, 0.3473, 1.5321, 1.5321, 2 11.5176 

10 -2, -1.6180, -1.6180, -0.6180, -0.6180, 0.6180, 0.6180, 1.6180, 

1.6180, 2 

12.944 

11 -1.9190, -1.9190, -1.3097, -1.3097, -0.2846, -0.2846, 0.8308, 

0.8308, 1.6825, 1.6825, 2 

14.0532 

 

12 -2, -1.732, -1.732, -1, -1, 0, 0, 1, 1, 1.732, 1.732, 2 14.928 

15 -1.9563, -1.9563, -1.6180, -1.6180, -1, -1, -0.2091, -0.2091, 

0.6180, 0.6180, 1.3383, 1.3383, 1.8271, 1.8271, 2 

19.1336 

16 -2, -1.8478, -1.8478, -1.4142, -1.4142, -0.7654, -0.7654, 0, 0, 

0.7654, 0.7654, 1.4142, 1.4142, 1.8478, 1.8478, 2 

20.1096 

 

Conclusion 

The characteristic polynomial in the above tables is easy to compute. When working with 

matrices that have complex expressions with parameters, which are frequently encountered in 

different mathematical models, the actual usefulness of this idea becomes evident. By 

expanding to first-order three-dimensional discrete dynamics, this method is useful in 

determining stability criteria for first-order two-dimensional discrete dynamics. Building upon 

these findings, our future work will expand these investigations to derive stability criteria for 

first-order four-dimensional discrete dynamics and, more generally, for first-order n-

dimensional discrete dynamics. 

 

 

 



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