Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

530 
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Quasi- Laplacian energy of some novel classes of graphs 

Manash Protim Borah a,*, Karam Ratan Singh b 

a,*  Department of Mathematics, L.T.K. College, Azad, North Lakhimpur, 787031, Assam, India, 
b Department of Basic and Applied Science, National Institute of  Technology Arunachal Pradesh, Papum pare, 791113, 

Arunachal Pradesh, India. 

 *Corresponding author 

Email address: mpborah36@gmail.com (Manash Protim Borah) 

 

Article History: 

Received: 26-10-2024 

Revised:10-11-2024 

Accepted:18-12-2024 

Abstract:  

We formulate the relationship of quasi-Laplacian energy of some novel classes of 

graphs with their corresponding original graphs. The novel graphs in our discussion 

are the 𝒮-graph, ℛ-graph, 𝒬-graph, total graph, and their join and corona operations 

graphs. The whole formulation is based on the relationship between quasi-Laplacian 

energy and the vertex degrees of the novel graph. It is also noted that quasi-

Laplacian energy is closely related with the first Zagreb index, number of vertices 

and edges of the graph. The exact formulas of quasi-Laplacian energy of novel 

graphs are obtained in terms of the corresponding quasi-Laplacian energies, the first 

Zagreb indices, and the number of vertices and edges of the original graphs. 

Keywords: Quasi-Laplacian energy, degree of vertex, Zagreb index, join, corona. 

     Mathematics Subject Classification: 05C07, 05C09, 05C50, 05C76 

 

1. Introduction 

All graphs discussed in this paper are simple and undirected. Let 𝐺 be a graph with vertices 

denoted by 𝑉(𝐺) and edges denoted by 𝐸(𝐺). Let |𝑉(𝐺)| = 𝑝 and |𝐸(𝐺)| = 𝑞.  

Let 𝑑𝐺(𝑣) represent the degree of vertices in 𝐺, where 𝑢 ∈ 𝑉(𝐺). Let 𝐷(𝐺) denote the diagonal 

matrix and 𝐴(𝐺) the adjacency matrix. The quasi Laplacian matrix, represented by 𝑄(𝐺), is 

defined as 𝑄(𝐺) = 𝐷(𝐺) + 𝐴(𝐺). Let 𝜇1(𝐺) ≥ 𝜇2(𝐺) ≥ ⋯ ≥ 𝜇𝑛(𝐺) denote the real, symmetric, 

and positive semi definite eigenvalues of 𝑄(𝐺). 

It is known that various graph operations can create novel classes of graphs from the original 

graphs. Therefore, understanding the relationships between some invariants of such novel 

graphs and the equivalent invariants of the original graphs is relevant. 

 

Graph energy is one such invariant based on the graph spectrum introduced by Gutman [9].  

We discuss here the quasi-Laplacian energy [6] of graph 𝐺, represented as 𝐸𝑄(𝐺) and 

determined by the equation 

𝐸𝑄(𝐺) =∑  

𝑝

𝑖=1

𝜇𝑖
2 



Communications on Applied Nonlinear Analysis 

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Yue, Cao and Qi, in the paper [6], defined quasi-Laplacian energy of 𝐺  is expressed as follows 

𝐸𝑄(𝐺) =∑  

𝑝

𝑖=1

 𝑑𝐺
2(𝑢𝑖) +∑  

𝑝

𝑖=1

 𝑑𝐺(𝑢𝑖), 𝑢 ∈ 𝑉(𝐺)………(1.1)  

Also,  equation (1.1) can be represented as follows 

𝐸𝑄(𝐺) =∑  

𝑝

𝑖=1

 𝑑𝐺
2(𝑣𝑖) +∑  

𝑝

𝑖=1

 𝑑𝐺(𝑣𝑖) = 𝑀1(𝐺) + 2𝑞 ………(1.2)  

where, 𝑀1 is called first Zagreb index [10] and second part of equation (1.2) is the handshaking 

lemma ∑𝑖=1
𝑝
 𝑑𝐺(𝑣𝑖) = 2𝑞. It is noticed that the quasi-Laplacian energy of any graph depends 

upon the degrees of vertices of that graph. 

Based on the above result discussed in the paper [6].  We formulate the relations of the quasi-

Laplacian energy of some classes of novel graphs in terms of corresponding original graphs. 

The subdivision graph 𝒮(𝐺), ℛ-graph ℛ(𝐺), 𝒬 graph 𝒬(𝐺), total graph 𝒯(𝐺) [3] and their join 

and corona operations graphs are the novel graphs for our discussion. Indulal [8] first defined 

𝒮-vertex and 𝒮 edge join and Liu and Zhang [16] determined their spectra. Lu and Miao [13] 

defined 𝒮-vertex corona and edge corona. Liu and Lu [14] defined 𝒮-vertex and edge 

neighbourhood corona.  Sun, Shang, and Bu [16] defined 𝒬-vertex and edge join graphs. The 𝒬-

vertex and edge corona are defined by Najiya and Chithra [2]. The definition of the corona of 

the total graph of one regular and another arbitrary graph, and spectra of the corona of the 

total graph is determined by Zhu, Tian, and Cui [19].  Also, 𝒮 vertex- ℛ vertex join,  𝒮 edge- ℛ 

edge join,  𝒮 vertex- ℛ edge join and 𝒮 edge- ℛ vertex join are defined by Das and Panigrahi [5, 

7]. The 𝒮-vertex-vertex-edge join of three 𝒮 graphs is defined by Wen, Zhang, and Li [17]. 

Berberler [1] derived the quasi-Laplacian energy of graphs based on ℛ graphs [13]. 

We derive formulas of the quasi-Laplacian energy of 𝒮,𝒬 and 𝒯 graphs and further, obtain 

quasi-Laplacian energy of  𝒮-vertex and edge join,  𝒮-vertex and edge corona, 𝒮-vertex and 

edge neighbourhood corona in terms of their corresponding original graphs. Similarly, we 

derive quasi-Laplacian energy of 𝒬-vertex and edge join, 𝒬-vertex and edge corona and 𝒯-

graph corona. The quasi-Laplacian energy of 𝒮 vertex- ℛ vertex join, 𝒮edge- ℛ edge join, 𝒮 

vertex- ℛ edge join, 𝒮 edge- ℛ vertex join and 𝒮-vertex-vertex-edge of three graphs join are 

also derived in terms of their corresponding original graphs. 



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Let 𝐺 be the (𝑝, 𝑞) graph where sets of old vertices |𝑉(𝐺)| = 𝑝 and sets of inserted new vertices 

𝐼(𝐺) ∣= 𝑞. Let 𝐻1 be the (𝑝1, 𝑞1) graph and 𝐻2 be the (𝑝2, 𝑞2) graph. Consequently, |𝑉(𝐻1)| =

𝑝1, |𝐼(𝐻1)| = 𝑞1, |𝑉(𝐻2)| = 𝑝2 and |𝐼(𝐻2)| = 𝑞2. 

2. 𝐸𝑄  based on 𝒮 graphs 

The quasi-Laplacian energy of the 𝒮 graph, 𝒮 join and 𝒮 corona graphs are formulated in this 

section based on the degree of vertices of the 𝒮-graph, 𝒮-join, and 𝒮-corona graphs 

respectively. Let 𝑢 be any vertex in 𝒮(𝐺), then the degree of 𝒮(𝐺) is represented by 

𝑑𝒮(𝐺)(𝑢𝑖) = {
𝑑𝐺(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐺);

2,  if 𝑢 ∈ 𝐼(𝐺).
 

Theorem 2.1. 𝐸𝑄(𝒮(𝐺)) = 𝐸𝑄(𝐺) + 6𝑞 

Proof. Using definition (1.1) on the 𝒮(𝐺)-graph, we have 

𝐸𝑄(𝒮(𝐺))= ∑  

|𝑉(𝒮(𝐺))|

𝑖=1

 𝑑𝒮(𝐺)
2 (𝑢𝑖) + ∑  

|𝑉(𝒮(𝐺))|

𝑖=1

 𝑑𝒮(𝐺)(𝑢𝑖)

=∑  

𝑝

𝑖=1

 𝑑(𝐺)
2 (𝑢𝑖) +∑  

𝑞

𝑖=1

 22 +∑ 

𝑝

𝑖=1

 𝑑(𝐺)(𝑢𝑖) +∑  

𝑞

𝑖=1

 2

= 𝐸𝑄(𝐺) + 6𝑞

 

Now, let 𝑢 be any vertex in 𝒮-vertex join (𝐻1 ∨̇ 𝐻2) of two graphs 𝐻1 and 𝐻2. Then, the degree of 

the vertex 𝑢 in (𝐻1 ∨̇ 𝐻2) is given by 

𝑑𝐻1V̇𝐻2(𝑢𝑖) = {

𝑑𝐻1(𝑢𝑖) + 𝑝2,  if 𝑢 ∈ 𝑉(𝐻1), for i = 1,2,… p1;

2,  if 𝑣 ∈ 𝐼(𝐺1);

𝑑𝐻2(𝑢𝑗) + 𝑝1,  if 𝑣 ∈ 𝑉(𝐻2), for j = 1,2,… p2.

 

and 𝐻1 ∨̇ 𝐻2 has 𝑝1 + 𝑞1 + 𝑝2 vertices. We derive quasi-Laplacian energy of 𝒮-vertex join in 

terms of corresponding original graphs 𝐻1 and 𝐻2. 

Theorem 2.2. 𝐸𝑄(𝐻1 ∨̇ 𝐻2) = 𝐸𝑄(𝐻1) + 𝐸𝑄(𝐻2) + 4𝑞1𝑝2 + 𝑝1𝑝2
2 + 6𝑞1 + 4𝑝1𝑞2 + 𝑝1

2𝑝2 + 2𝑝1𝑝2 

Proof. Using definition (1.1), quasi-Laplacian energy of 𝒮-vertex join is obtained by 



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𝐸𝑄(𝐻1 ∨̇ 𝐻2) = ∑  

𝑝1+𝑞1+𝑝2

𝑖=1

 𝑑𝐻1∨̇𝐺2
2 (𝑢𝑖) + ∑  

𝑝1+𝑞1+𝑝2

𝑖=1

 𝑑𝐻1∨̇𝐻2(𝑢𝑖)

=∑  

𝑝1

𝑖=1

 𝑑(𝐻1)
2 (𝑢𝑖) + 2𝑝2∑ 

𝑝1

𝑖=1

 𝑑(𝐻1)(𝑢𝑖) + 𝑝2
2∑ 

𝑝1

𝑖=1

 + 4𝑞1

+∑ 

𝑝2

𝑗=1

 𝑑(𝐻2)
2 (𝑢𝑗) + 2𝑝1∑ 

𝑝2

𝑗=1

 𝑑(𝐻2)(𝑢𝑗) + 𝑝1
2∑ 

𝑝2

𝑗=1

 

+∑  

𝑝1

𝑖=1

 𝑑(𝐻1)(𝑢𝑖) + 𝑝2∑ 

𝑝1

𝑖=1

 + 2𝑞1 +∑  

𝑝2

𝑗=1

 𝑑(𝐻2)(𝑢𝑗) + 𝑝1∑ 

𝑝2

𝑗=1

 

= 𝐸𝑄(𝐻1) + 𝐸𝑄(𝐻2) + 4𝑞1𝑝2 + 𝑝1𝑝2
2 + 6𝑞1 + 4𝑝1𝑞2 + 𝑝1

2𝑝2 + 2𝑝1𝑝2.

 

Next, we formulate a relation of quasi-Laplacian energy of 𝐻1 ∨ 𝐻2  join. 

Let 𝑢 be any vertex in 𝒮-edge join. The degree of vertex 𝑢 in 𝐻1 ∨ 𝐻2 join is given by 

𝑑𝐻1∪𝐻2(𝑢𝑖) = {

𝑑𝐻1(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻1);

2 + 𝑝2,  if 𝑢 ∈ 𝐼(𝐻1)

𝑑𝐻2(𝑣𝑖) + 𝑞1,  if 𝑢 ∈ 𝑉(𝐻2).

; 

and 𝐻1 ∨ 𝐻2 has 𝑝1 + 𝑞1 + 𝑝2 vertices. 

Theorem 2.3. 𝐸𝑄(𝐻1 ∨ 𝐻2) = 𝐸𝑄(𝐻1) + 𝐸𝑄(𝐻2) + 6𝑞1𝑝2 + 𝑞1𝑝2
2 + 6𝑞1 + 4𝑞1𝑞2 + 𝑞1

2𝑝2 

Proof. Using equation (1.1), we get 

𝐸𝑄(𝐻1 ∨ 𝐻2) = ∑  

𝑝1+𝑞1+𝑝2

𝑖=1

 𝑑𝐻1⊵𝐻2
2 (𝑢𝑖) + ∑  

𝑝1+𝑞1+𝑝2

𝑖=1

 𝑑𝐻1∨𝐻2(𝑢𝑖)

=∑  

𝑝1

𝑖=1

 𝑑𝐻1
2 (𝑢𝑖) +∑  

𝑞1

𝑖=1

  (2 + 𝑝2)
2 +∑ 

𝑝2

𝑖=1

  (𝑑(𝐻2)(𝑢𝑖) + 𝑞1)
2

+∑ 

𝑝1

𝑖=1

 𝑑(𝐻1)(𝑢𝑖) +∑  

𝑞1

𝑖=1

 2 +∑  

𝑝2

𝑖=1

 𝑑(𝐻2)(𝑢𝑖) + 𝑞1

 

The result can be derived easily. 

2.1. 𝐸𝑄  of 𝒮-vertex and edge corona 

Let 𝑢 be any vertex in 𝒮-vertex corona. The degree of vertex of 𝑢 in 𝒮-vertex corona is given by 

𝑑𝐻1⊙𝐻2(𝑢𝑖) = {

𝑑𝐻1(𝑢𝑖) + 𝑝2,  if 𝑢 ∈ 𝑉(𝐻1);

2,  if 𝑢 ∈ 𝐼(𝐻1);

𝑑𝐻2(𝑣𝑗
𝑖) + 1,  if 𝑢 = 𝑣𝑗

𝑖 , for 𝑖 = 1,2,… 𝑝1, for 𝑗 = 1,2,… 𝑝2.

 

and 𝐻1⊙𝐻2 has 𝑝1 + 𝑞1 + 𝑝1𝑝2 vertices. The formula of quasi-Laplacian energy of 𝒮-vertex 
corona is given below 



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Theorem2.4. 𝐸𝑄(𝐻1⊙𝐻2) = 𝐸𝑄(𝐻1) + 𝑝1𝐸𝑄(𝐻2) + 𝑝1𝑝2
2 + 4𝑞1𝑝2 + 3𝑞1 + 3𝑝1𝑝2 + 4𝑝1𝑞2 

Proof.  By using definition (1.1), we get 

𝐸𝑄(𝐻1⊙𝐻2) = ∑  

𝑝1+𝑞1+𝑝1𝑝2

𝑖=1

 𝑑𝐻1⊙𝐻2
2 (𝑣𝑖) + ∑  

𝑝1+𝑞1+𝑝1𝑝2

𝑖=1

 𝑑𝐻1⊙𝐻2(𝑢𝑖)

=∑  

𝑝1

𝑖=1

 𝑑𝐻1(𝑢𝑖) + 𝑝2)

2

+∑ 

𝑞1

𝑖=1

 22 +∑  

𝑝1

𝑖=1

 ∑  

𝑝2

𝑗=1

  (𝑑𝐻2(𝑣𝑗
𝑖) + 1)

2

+∑  

𝑝1

𝑖=1

 𝑑(𝐻1)(𝑢𝑖) + 𝑝2 +∑ 

𝑞1

𝑖=1

 2 +∑  

𝑝1

𝑖=1

 ∑  

𝑝2

𝑗=1

 𝑑𝐻2(𝑣𝑗
𝑖) + 1

= 𝑝2
2∑ 

𝑝1

𝑖=1

 + 2𝑝2∑ 

𝑝1

𝑖=1

 𝑑𝐻1(𝑢𝑖) +∑  

𝑝1

𝑖=1

 𝑑𝐻1
2 (𝑢𝑖) + 4𝑞1 +∑  

𝑝1

𝑖=1

 ∑  

𝑝2

𝑗=1

 

+2∑  

𝑝1

𝑖=1

 ∑  

𝑝2

𝑗=1

 𝑑𝐻2(𝑣𝑗) +∑  

𝑝1

𝑖=1

 ∑  

𝑝2

𝑗=1

 𝑑𝐻2
2 (𝑣𝑗) + 𝑝1𝑝2 +∑  

𝑝1

𝑖=1

 𝑑𝐻1(𝑢𝑖)

+2𝑞1 +∑ 

𝑝1

𝑖=1

 ∑  

𝑝2

𝑗=1

  +∑  

𝑝1

𝑖=1

 ∑  

𝑝2

𝑗=1

 𝑑𝐻2(𝑣𝑗)

 

Hence, the result follows. 

Let 𝑢 be any vertex in 𝒮-edge corona, then the degree of the vertex 𝑢 is given by 

𝑑𝐻1⊖𝐻2(𝑢𝑖) = {

𝑑𝐻1(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻1);

2 + 𝑝2,  if 𝑢 ∈ 𝐼(𝐻1)

𝑑𝐻2(𝑣𝑗
𝑖) + 1,  if 𝑢 = 𝑣𝑗

𝑖 , for 𝑖 = 1,2,… 𝑞1, for 𝑗 = 1,2,… 𝑝2.

; 

and 𝐻1Θ𝐻2 has 𝑝1 + 𝑞1 + 𝑞1𝑝2 vertices. The formula of quasi-Laplacian energy of 𝒮-edge 

corona is given by 

Theorem 2.5. 𝐸𝑄(𝐻1⊖𝐻2) = 𝐸𝑄(𝐻1) + 𝑞1𝐸𝑄(𝐻2) + 6𝑞1 + 7𝑞1𝑝2 + 4𝑞1𝑞2 

Proof.  Using definition (1.1), 



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𝐸𝑄(𝐻1⊙𝐻2) = ∑  

|𝑉(𝐻1⊖𝐻2)|

𝑖=1

 𝑑𝐻1⊖𝐻2
2 (𝑢𝑖) + ∑  

|𝑉(𝐻1⊖𝐻2)|

𝑖=1

 𝑑𝐻1⊖𝐻2(𝑢𝑖)

= ∑  

𝑝1+𝑞1+𝑞1𝑝2

𝑖=1

 𝑑𝐻1⊖𝐻2
2 (𝑢𝑖) + ∑  

𝑝1+𝑞1+𝑞1𝑝2

𝑖=1

 𝑑𝐻1⊖𝐻2
(𝑢𝑖)

=∑  

𝑝1

𝑖=1

 𝑑𝐻1
2 (𝑢𝑖) +∑  

𝑞1

𝑖=1

  (2 + 𝑝2)
2 +∑  

𝑞1

𝑖=1

 ∑  

𝑝2

𝑗=1

  (𝑑𝐻2(𝑣𝑗
𝑖) + 1)

2

+∑  

𝑝1

𝑖=1

 𝑑(𝐻1)(𝑢𝑖) +∑  

𝑞1

𝑖=1

 2 +∑  

𝑞1

𝑖=1

 ∑  

𝑝2

𝑗=1

 𝑑𝐻2(𝑣𝑗
𝑖) + 1

 

Hence, the result follows. 

2.2. 𝐸𝑄  of 𝒮-vertex and edge neighbourhood corona 

Let 𝑢 be any vertex in 𝒮-vertex neighbourhood corona. Then degree of vertices is given by 

𝑑𝐻1⊟𝐻2(𝑣𝑖) = {

𝑑𝐻1(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻1);

2 + 2𝑝2,  if 𝑢 ∈ 𝐼(𝐻1);

𝑑𝐻1(𝑢𝑖) + 𝑑𝐻2(𝑣𝑗),  if 𝑢 = 𝑣𝑗
𝑖 , for 𝑖 = 1,2,… 𝑝1, for 𝑗 = 1,2,… 𝑝2.

 

and 𝐻1⊟𝐻2 has 𝑝1 + 𝑞1 + 𝑝1𝑝2 vertices. The quasi-Laplacian energy of 𝒮-vertex 

neighbourhood corona is given by 

Theorem 2.6. 𝐸𝑄(𝐻1⊟𝐻2) = (1 + 𝑝2)𝐸𝑄(𝐻1) + 𝑝1𝐸𝑄(𝐻2) + 6𝑞1 + 10𝑞1𝑝2 + 4𝑞1𝑝2
2 + 8𝑞1𝑞2 

Proof: Using definition (1.1), the above Theorem can be easily proved. 

Also, the degree of any vertex of 𝑢 in 𝒮-edge neighbourhood corona is given by 

𝑑𝐻1⊟𝐻2(𝑢𝑖) = {

(1 + 𝑝2)𝑑𝐻1(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻1);

2,  if 𝑢 ∈ 𝐼(𝐻1)

2 + 𝑑𝐻2(𝑢𝑗),  if 𝑢 = 𝑣𝑗
𝑖 , for 𝑖 = 1,2,… 𝑞1, for 𝑗 = 1,2, … 𝑝2.

; 

and 𝐻1⊟𝐻2 has 𝑝1 + 𝑞1 + 𝑞1𝑝2  vertices. The quasi-Laplacian energy of 𝒮-edge neighbourhood 

corona is given by 

Theorem 2.7. 𝐸𝑄(𝐻1⊟𝐻2) = (1 + 2𝑝2)𝐸𝑄(𝐻1) + 2𝑞1𝐸𝑄(𝐻2) + 𝑝2
2𝑀1(𝐻1) + 4𝑞1 + 2𝑞1𝑝2 +

12𝑞1𝑞2, where 𝑀1 is the first Zagreb index. 

Using definition (1.1), the above Theorem can be easily proved. 

 



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3. 𝐸𝑄  based on 𝒬 graph 

The energy of the 𝒬 graph, 𝒬 join and 𝒬 corona graphs are formulated in this section based on 

the degree of vertices of the 𝒬-graph, 𝒬-join, and 𝒬-corona graphs respectively. 

Let 𝑢 be any vertex in 𝒬(𝐺). Then, the degree of vertex is given by 

𝑑𝒮(𝐺)(𝑢𝑖) = {
𝑑𝐺(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐺);

4,  if 𝑢 ∈ 𝐼(𝐺).
 

We derive a formula of energy of 𝒬(𝐺). 

Theorem 3.1. 𝐸𝑄(𝒬(𝐺)) = 𝐸𝑄(𝐺) + 20𝑞 

Proof. We follow from definition (1.1), 

𝐸𝑄(𝒬(𝐺)) == ∑  

𝑝+𝑞

𝑖=1

 𝑑𝒬(𝐺)
2 (𝑣𝑖) +∑  

𝑝+𝑞

𝑖=1

 𝑑𝒬(𝐺)(𝑢𝑖)

= ∑  

𝑝

𝑖=1

 𝑑(𝐺)
2 (𝑢𝑖) +∑  

𝑞

𝑖=1

 42 +∑  

𝑝

𝑖=1

 𝑑(𝐺)(𝑢𝑖) +∑  

𝑞

𝑖=1

 4

= 𝐸𝑄(𝐺) + 20𝑞

 

3.1. 𝐸𝑄  of 𝒬-vertex and edge join 

Let 𝑢 be any vertex in 𝐻1⟨𝑣⟩𝐻2. Then, the degree of vertex of 𝐻1⟨𝑣⟩𝐻2  is given by 

𝑑𝐻1⟨𝑣⟩𝐻2(𝑢𝑖) = {

𝑑𝐻1(𝑢𝑖) + 𝑝2,  if 𝑢 ∈ 𝑉(𝐻1);

4,  if 𝑢 ∈ 𝐼(𝐻1);

𝑑𝐻2(𝑢𝑖) + 𝑝1,  if 𝑢 ∈ 𝑉(𝐻2).

 

𝐻1⟨𝑣⟩𝐻2 has 𝑝1 + 𝑞1 + 𝑝2 vertices. The relation of quasi-Laplacian energy in between 𝒬-vertex 

join with corresponding two original graphs is obtained easily by using definition (1.1) as 

follows 

Theorem  3.2. 𝐸𝑄(𝐻1⟨𝑣⟩𝐻2) = 𝐸𝑄(𝐻1) + 𝐸𝑄(𝐻2) + 4𝑞1𝑝2 + 𝑝1𝑝2
2 + 20𝑞1 + 4𝑝1𝑞2 + 𝑝1

2𝑝2 +

2𝑝1𝑞2 

Also, let 𝑢 be any vertex in 𝒬-edge join. The degree of vertex 𝑢 of 𝒬-edge join is given by 

𝑑𝐻1⟨𝑒⟩𝐻2(𝑢𝑖) = {

𝑑𝐻1(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻1);

2 + 𝑝2,  if 𝑢 ∈ 𝐼(𝐻1)

𝑑𝐻2(𝑢𝑖) + 𝑞1,  if 𝑢 ∈ 𝑉(𝐻2).

; 



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𝐻1⟨𝑒⟩𝐻2 has 𝑝1 + 𝑞1 + 𝑝2 vertices. The relation of quasi-Laplacian energy in between 𝒬-edge 

join with corresponding two original graphs is obtained easily by using definition (1.1) as 

follows 

Theorem  3.3. 𝐸𝑄(𝐻1⟨𝑒⟩𝐻2) = 𝐸𝑄(𝐻1) + 𝐸𝑄(𝐻2) + 12𝑞1𝑝2 + 𝑞1𝑝2 + 20𝑞1 + 𝑞1
2𝑝2 + 𝑞1𝑝2

2 

3.2. 𝐸𝑄  of  𝒬-vertex and edge corona 

Let 𝑢 be any vertex in 𝒬-vertex corona. Then, the degree of any vertex 𝑢 in 𝒬-vertex corona is 

given by 

𝑑𝐻1O𝐻2(𝑣𝑖) = {

𝑑𝐻1(𝑢𝑖)+2,  if 𝑢 ∈ 𝑉(𝐻1);

4,  if 𝑢 ∈ 𝐼(𝐻1)

𝑑𝐻2(𝑣𝑗
𝑖) + 1,  if 𝑢 = 𝑣𝑗

𝑖 , for 𝑖 = 1,2,3, …𝑝1, for 𝑗 = 1,2,3, . . 𝑝2.

; 

𝐻1o𝐻2 has 𝑝1 + 𝑞1 + 𝑝1𝑝2 vertices. Then, by using equation (1.1) we get 

 
Theorem 3.4. 𝐸𝑄(𝐻1𝑜𝐻2) = 𝐸𝑄(𝐻1) + 𝑝1𝐸𝑄(𝐻2) + 𝑝1𝑝2

2 + 20𝑞1 + 4𝑝1𝑞2 + 3𝑝1𝑝2 + 4𝑞1𝑝2 

Next, the degree of any vertex of 𝑢 in 𝒬-edge corona is given by 

𝑑𝐻1⊛𝐻2(𝑢𝑖) = {

𝑑𝐻1(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻1);

𝑝2 + 4,  if 𝑢 ∈ 𝐼(𝐻1);

𝑑𝐻2(𝑣𝑗
𝑖) + 1,  if 𝑢 = 𝑣𝑗

𝑖 , for 𝑖 = 1,2, … 𝑝1, for 𝑗 = 1,2,… 𝑝2.

 

𝐺1⊛𝐺2 has 𝑝1 + 𝑞1 + 𝑞1𝑝2  vertices. Then, the quasi-Laplacian energy of 𝒬-edge corona is 

easily obtained as follows 

Theorem 3.5.   𝐸𝑄(𝐻1⊛𝐻2) = 𝐸𝑄(𝐻1) + 𝑝1𝐸𝑄(𝐻2) + 𝑞1𝑝2
2 + 9𝑞1𝑝2 + 20𝑞1 + 4𝑞2𝑝1 + 2𝑝1𝑞2 

4. 𝐸𝑄  based on 𝒯-graph 

We derive energy of 𝒯 graph and 𝒯 graph corona based on the degree of vertices of the 𝒯-

graph and 𝒯-corona graphs are determined here. 

Let 𝑢 be any vertex in 𝒯(𝐺), then the degree of vertex 𝑢 in 𝒯(𝐺) is given by 

𝑑𝒯(𝐺)(𝑢𝑖) = {
2𝑑𝐺(𝑣𝑖),  if 𝑢 ∈ 𝑉(𝐺);

4,  if 𝑢 ∈ 𝐼(𝐺).
 

The vertices of 𝒯(𝐺) is 𝑝 + 𝑞. The 𝑄 energy of 𝒯(𝐺) is given by 

Theorem 4.1. 𝐸𝑄(𝒯(𝐺)) = 2𝐸𝑄(𝐺) + 20𝑞 + 2𝑀1(𝐺) 

Where, 𝑀1 is the first Zagreb index. 



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4.1. 𝐸𝑄  of 𝒯-graph corona 

Let 𝐻1 be 𝑟1 regular and 𝐻2 be any graph. Also, let 𝑢 be any vertex in 𝒯(𝐺). Then, the degree of 

vertex of 𝒯 - corona is given by 

𝑑𝐻1⋆𝐻2(𝑢𝑖) = {

2𝑑𝐻1(𝑢𝑖) + 𝑝2,  if 𝑢 ∈ 𝑉(𝐻1);

2𝑟1,  if 𝑢 ∈ 𝐼(𝐻1)

𝑑𝐻2(𝑣𝑗
𝑖) + 1,  if 𝑢 = 𝑣𝑗

𝑖 , for 𝑖 = 1,2,… 𝑝1, for 𝑗 = 1,2,… 𝑝2.

; 

𝐻1 ⋆ 𝐻2 has 𝑝1 + 𝑞1 + 𝑝1𝑝2 vertices. The 𝐸𝑄  of 𝒯-graph corona is easily obtained by using 

equation (1.1) as follows 

Theorem 4.2. 𝐸𝑄(𝐻1 ⋆ 𝐻2) = 2𝐸𝑄(𝐻1) + 𝑝1𝐸𝑄(𝐻2) + 2𝑀1(𝐻1) + 5𝑝1𝑝2 + 𝑝2
2𝑝1 + 4𝑟1

2𝑞1 +

2𝑟1𝑞1 + 4𝑝1𝑞2 + 2𝑝1𝑝2 
where, 𝑀1 is the first Zagreb index. 

5. 𝐸𝑄  based on 𝒮-graph and ℛ-graph join 

The 𝐸𝑄  energy of four 𝒮-graph and ℛ-graph joins with their corresponding original graphs 𝐻1 

and 𝐻2 are formulated in this part. 

5.1. 𝐸𝑄  of 𝒮-vertex and ℛ-vertex join 

First, we formulate the quasi-Laplacian energy of 𝒮-vertex and ℛ-vertex join. Let 𝑢 be any 

vertex in 𝒮-vertex and ℛ-vertex join. Then, the degrre of any vertex in 𝒮(𝐻1) ∨̈ ℛ(𝐻2) is given 

by 

𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖) = {

𝑑𝐻1(𝑢𝑖) + 𝑝2,  if 𝑢 ∈ 𝑉(𝐻1);

2,  if 𝑢 ∈ 𝐼(𝐻1) ∪ 𝐼(𝐻2);

2𝑑𝐻2(𝑢𝑖) + 𝑝1,  if 𝑢 ∈ 𝑉(𝐻2).

 

𝒮(𝐻1) ∨̈ ℛ(𝐻2) has 𝑝1 + 𝑞1 + 𝑝2 + 𝑞2 vertices. 

Theorem 5.1. The quasi-Laplacian energy of 𝐸𝑄(𝒮(𝐻1) ∨̈ ℛ(𝐻2)) = 𝐸𝑄(𝐻1) + 4𝐸𝑄(𝐻2) + 𝑝2
2𝑝1 +

𝑝1
2𝑝2 + 4𝑞1𝑝2 + 8𝑞2𝑝1 + 2𝑞2 + 6𝑞1 + 2𝑝1𝑝2 

Proof.  By using equation (1.1), we get 



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𝐸𝑄(𝒮(𝐻1) ∨̈ ℛ(𝐻2)) = ∑  

𝑝1+𝑞1+𝑝2+𝑞2

𝑖=1

 𝑑𝒮(𝐻1)
2 ∨̈ ℛ(𝐻2)(𝑢𝑖) + ∑  

𝑝1+𝑞1+𝑝2+𝑞2

𝑖=1

 𝑑𝒮(𝐺1)∨̈ℛ(𝐺2)(𝑢𝑖)

=∑  

𝑝1

𝑖=1

 𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)
2 (𝑢𝑖) +∑  

𝑞1

𝑖=1

 𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)
2 (𝑢𝑖) +∑  

𝑝2

𝑗=1

 𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)
2 (𝑢𝑖)

+∑  

𝑞2

𝑖=1

 𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)
2 (𝑢𝑖) +∑  

𝑝1

𝑖=1

 𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖) +∑  

𝑞1

𝑖=1

 𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖)

+∑  

𝑝2

𝑖=1

 𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖) +∑  

𝑞2

𝑖=1

 𝑑𝒮(𝐻1)∨̈ℛ(𝐻2)(𝑢𝑖)

 

Hence,  the result follows. 

5.2. 𝐸𝑄  of  𝒮-edge and  ℛ-edge join 

Let 𝑢 be any vertex in 𝒮-edge and ℛ-edge join. Then, the degree of any vertex of 𝑢 in 𝒮-edge 

and ℛ-edge join 𝒮(𝐻1)𝑉‾ℛ(𝐻2) is given by 

𝑑𝒮(𝐻1)𝑉‾ℛ(𝐻2)(𝑢𝑖) =

{
 
 

 
 𝑑𝐻1(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻1);

2 + 𝑞2,  if 𝑣 ∈ 𝐼(𝐻1);

2𝑑𝐻2(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻2);

2 + 𝑞1,  if 𝑢 ∈ 𝐼(𝐻2).

 

𝒮(𝐻1)𝑉‾ℛ(𝐻2) has 𝑝1 + 𝑞1 + 𝑝2 + 𝑞2 vertices. The 𝑄-energy is obtained easily by using equation 

(1.1). 

Theorem  5.2. 𝐸𝑄 (𝒮(𝐻1)𝑉‾ℛ(𝐻2)) = 𝐸𝑄(𝐻1) + 4𝐸𝑄(𝐻2) + 6𝑞1 + 10𝑞1𝑞2 + 𝑞1𝑞2(𝑞1 +𝑞2) + 6𝑞2 

5.3. 𝐸𝑄  of  𝒮-vertex and  ℛ-edge join 

Let 𝑢 be any vertex in 𝒮-vertex and ℛ-edge join. Then, the degree of any vertex of 𝑢 in 𝒮-vertex 

and ℛ-edge join 𝒮(𝐻1)𝑉‾̇ℛ(𝐻2) is given by 

𝑑𝒮(𝐻1)�̇�ℛ(𝐻2)(𝑢𝑖) =

{
 
 

 
 𝑑𝐻1(𝑢𝑖) + 𝑞2,  if 𝑢 ∈ 𝑉(𝐻1);

2,  if 𝑢 ∈ 𝐼(𝐻1);

2𝑑𝐻2(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻2);

2 + 𝑝1,  if 𝑢 ∈ 𝐼(𝐻2).

 

𝒮(𝐺1)𝑉‾̇ℛ(𝐺2) has 𝑝1 + 𝑞1 + 𝑝2 + 𝑞2 vertices. We get the following result by using equation 

(1.1). 



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Theorem 5.3. 𝐸𝑄 (𝒮(𝐻1)𝑉‾̇ℛ(𝐻2)) = 𝐸𝑄(𝐻1) + 4𝐸𝑄(𝐻2) + 𝑞2
2𝑝1 + 4𝑞1𝑞2 + 6𝑞1 + 6𝑞2𝑝1 + 𝑝2

2𝑞2 +

4𝑞2 

5.4. 𝐸𝑄  of  𝒮-edge and  ℛ-vertex join 

Let 𝑢 be any vertex in 𝒮-edge and ℛ-vertex join. Then, the degree of any vertex of 𝑢 in 𝒮-edge 

and ℛ-vertex join 𝒮(𝐻1)𝑉‾̇ℛ(𝐻2) is given by 

𝑑𝒮(𝐻1)𝑉‾ℛ(𝐻2)(𝑢𝑖) =

{
 
 

 
 𝑑𝐻1(𝑢𝑖),  if 𝑢 ∈ 𝑉(𝐻1);

2 + 𝑝2,  if 𝑢 ∈ 𝐼(𝐻1);

2𝑑𝐻2(𝑢𝑖) + 𝑞1,  if 𝑢 ∈ 𝑉(𝐻2);

2,  if 𝑢 ∈ 𝐼(𝐻2).

 

𝒮(𝐻1)�̇�ℛ(𝐻2) has 𝑝1 + 𝑞1 + 𝑝2 + 𝑞2 vertices. We easily get the following Theorem by using 

equation (1.1) 

Theorem 5.4. 𝐸𝑄 (𝒮(𝐻1)�̇�ℛ(𝐻2)) = 𝐸𝑄(𝐻1) + 4𝐸𝑄(𝐻2) + 6𝑞1 + 6𝑞1𝑝2 + 𝑞1
2𝑝2 + 𝑞1𝑝2

2 + 10𝑞2 

6. 𝐸𝑄  of  𝒮-vertex-vertex-edge join of triple graphs 

Let 𝐻3 be (𝑝3, 𝑞3) graph. Then 𝒮-vertex-vertex-edge join of three 𝒮 graphs is denoted by 𝐻1
𝑆 ▹

(𝐻2
𝑉 ∪ 𝐻3

𝐸). The degree of vertex 𝑢 ∈ 𝐻1
𝑆 ▹ (𝐻2

𝑉 ∪ 𝐻3
𝐸) is given by 

𝑑𝐻1𝑆▹(𝐻2𝑉∪𝐻3𝐸)
(𝑢𝑖) =

{
 
 

 
 
𝑑𝐻1(𝑢𝑖) + 𝑝2,  if 𝑢 ∈ 𝑉(𝐻1);

2 + 𝑝1,  if 𝑢 ∈ 𝐼(𝐻1);

𝑑𝐻2(𝑢𝑗) + 𝑝1,  if 𝑢 ∈ 𝑉(𝐻2), 𝑗 = 1,2,… 𝑝2;

𝑑𝐻3(𝑤𝑘),  if 𝑢 = 𝑤𝑘 ∈ 𝑉(𝐻3), 𝑘 = 1,2…𝑝3.

 

𝐻1
𝑆 ▹ (𝐻2

𝑉 ∪ 𝐻3
𝐸) has 𝑝1 + 𝑞1 + 𝑝2 + 𝑝3  vertices. The following Theorem is easily obtained by 

using equation (1.1). 

Theorem 6.1. 𝐸𝑄(𝐻1
𝑆 ▹ (𝐻2

𝑉 ∪ 𝐻3
𝐸) = 𝐸𝑄(𝐻1) + 𝐸𝑄(𝐻2) + 𝐸𝑄(𝐻3) + 4𝑞1𝑝2 +𝑞1𝑝2

2 + 6𝑞1 +

5𝑞1𝑝1 + 𝑝1
2𝑞1 + 4𝑞2𝑝1 + 𝑝1

2𝑝2 + 4𝑞1𝑞3 + 𝑞1
2𝑝3 + 2𝑝1𝑝2 + 𝑞1𝑝3. 

Coclusion: 

We establish a relation between the quasi-Laplacian energy of few novel graphs and their 

respective original graphs. The formulation is predicated on the relationships between quasi 

Laplacian energy and vertex degree in the novel graphs. Moreover, it is directly associated with 

the first Zagreb index, quantity of vertices and edges of the graph. 



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