Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2s (2025) 

 

296 
 

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Even Vertex Odd Edge Root Square Mean Labelling of Path Related 

Graphs 

 

K. N. Babu1, S. Meenakshi2  
1 Research Scholar, Department of Mathematics, Vels Institute of Science, Technology and Advanced Studies (VISTAS), 

Associate Professor, Sri Malolan College of Arts and Science. Email: babumalolan@gmail.com  

2 Research Supervisor, Vels Institute of Science, Technology and Advanced Studies (VISTAS), Chennai, India. 

 Email: meenakshikarthikeyan@yahoo.in 

 

Article History: 

Received: 14-09-2024 

Revised: 22-10-2024 

Accepted: 02-11-2024 

Abstract:  

Consider G be a graph with p vertices and q edges. A graph G is said to be even vertex odd 

edge root square mean labelling if there exist an injective map f : V(G) → {0,1,2,3, … , 2q} 

such that the induced edge labels are odd and distinct which can be obtained by f
∗ (e) =

⌈√
f(u)2 +f(v)2

2
⌉  or  ⌊√

f(u)2 +f(v)2

2
⌋. Any graph which admits even vertex odd edge root square 

mean labelling then it is called as even vertex odd edge root square mean labelling graph.   

Keywords: EVOE – RSML, Graph, Ladder, Corona graph 

 

1.Introduction: 

Throughout the work, we consider a simple graph. The terminology and the definitions of graph were 

followed in [1]. For a complete analysis of labeling studied by Gallian [1]. In 1960’s graph labeling 

was introduced by Rosa. Odd vertex even edge root square mean labeling discussed in [5] which 

motivated to construct the above said labeling pattern. We study a few paths related graphs which 

satisfies the above pattern. Mean labeling, root square mean and super root square mean labeling were 

discussed in [4,6,7,8]. 

2 Preliminary: 

Definition 2.1 [5] 

 A graph G is said to be odd vertex even edge root square mean labelling if there exist an injective map 

f : V(G) → {1,2,3, … , 2q + 1} such that the induced edge labels are odd and distinct which can be 

obtained by f
∗ (e) = ⌈√f(u)2 +f(v)2

2
⌉  or  ⌊√f(u)2 +f(v)2

2
⌋ .  Any graph which admits even vertex odd edge 

root square mean labelling then it is called as odd vertex even edge root square mean labelling graph. 

 Definition 2.2 

A graph G is said to be even vertex odd edge root square mean labelling if there exist an injective map 

f : V(G) → {0,1,2,3, … , 2q} such that the induced edge labels are odd and distinct which can be obtained 

by f
∗ (e) = ⌈√f(u)2 +f(v)2

2
⌉  or  ⌊√f(u)2 +f(v)2

2
⌋ .  Any graph which admits even vertex odd edge root square 

mean labelling then it is called as even vertex odd edge root square mean labelling graph.  

mailto:babumalolan@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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 Definition 2.3 [6]:  

A graph G with p vertices and q edges, is a mean graph if there is an injective function f from the 

vertex set to { 1, 2, 3…q} when each edge uv is labelled with 
𝑓(𝑢)+𝑓(𝑣)

2
 if f(u)+f(v) is even and 

𝑓(𝑢)+𝑓(𝑣)+1

2
  if f(u)+f(v)+1 is odd then the labelling edges are distinct.  

Definition 2.4 [4]:  

A graph G with p vertices and q edges is called a Root Square Mean graph if is possible to label the 

vertices v with distinct labels f(x) from 1, 2, 3,… q+1 in such a way that when each edge e = uv is 

labelled with f(e ) =⌊√ (
f(u)2 +f(v)2

2
 )  ⌋ or ⌈√ (

f(u)2 +f(v)2

2
 ) ⌉ then each labels are distinct. 

 Definition 2.5 [4]:  

The corona graph is obtained by taking one copy of path Pn and n copies of K1 then joining the ith 

vertex of Pn with an edge to every vertex in the ith copy of K1. It is denoted by Pn K1.  

Definition 2.6 [7]:  

TW(Pn) is a graph which is obtained from a path by identifying k1,2 to all the vertices of the path 

except one pendent vertex (Twing graph) A path with a least vertex is connected and has two terminal 

vertices while all other vertices have degree 2. 

Theorem 3.1: A path Pn is an even vertex odd edge root square mean labelling graph. 

Proof:  

Let G be a Path Pn, where V(G)= {ui:1≤ 𝑖 ≤ 𝑛} and E(G) = {ui, ui+1: 1≤ 𝑖 ≤ 𝑛 − 1} It is evident 

that G allows an even-vertex odd-edge root square mean labeling, as labels can be assigned to its 

vertices and edges in a sequential manner. Hence, G is classified as an even-vertex odd-edge root 

square mean graph. 

Illustration: Figure 1 shows G = P7 is Even vertex odd edge root square mean labelling. 

 
Figure 1 Even vertex odd edge root square mean labelling of P7 

Theorem 3.2: Pn ʘ K1 is an even vertex odd edge root square mean labelling graph. 

Proof: Let G = Pn ʘ K1 

The vertex set and edge set of G is defined as follows; 

V(G) = {ui , u′i
  ; 1≤ 𝑖 ≤ 𝑛} and 

E(G) = {ui u′i
  ; 1≤ 𝑖 ≤ 𝑛 and ui u′i+1

  ; 1≤ 𝑖 ≤ 𝑛-1 } 

We define a map f: V(G) → {0,1, 2, 3…….2q}. 

         

       



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The labelling pattern of vertices can be defined as follows; 

f (u2i−1)  = 8i – 6  and  f (u2i)  = 8i – 4; 1≤ 𝑖 ≤ 𝑛 

f (u′2i−1) = 8(i-1)  and  f (u′2i)   = 8i-2; 1≤ 𝑖 ≤ 𝑛 

Then the induced edge labels are 

f ′ ( uiu′i) = 4i – 3;  1 ≤  i ≤  n  

f ′ ( uiu′i+1) = 4i - 1  ;  1 ≤  i ≤  n − 1  

Thus G = Pn ʘ K1 admit even vertex odd edge root square mean labelling graph. 

Illustration: Figure 2 illustrate the G = P4 ʘ K1 is even vertex odd edge root square mean labeling  

 
Figure 2 G = P4 ʘ K1 

Theorem 3.3: Pn ʘ K1,2 is an even vertex odd edge root square mean labelling graph. 

Proof: Let G = Pn ʘ K1,2 is a graph formed by attaching a K1,2 (a star with one central vertex and 

two leaves) to each vertex of the path Pn. 

 Let V(G) = {ui , u′i  , u′′i
   ; 1≤ 𝑖 ≤ 𝑛} and  

E(G) = {ui u′i  , ui u′′
i
   : 1≤ 𝑖 ≤ 𝑛 and ui  u′i+1  : 1≤ 𝑖 ≤ 𝑛-1}  

This describes the standard labeling of the vertices and edges of G. 

We define a map f: 𝑉(𝐺) → {0, 1, 2, ……2q} by  

f(𝑢𝑖) = 6𝑖 – 4 ∶ 𝑖 ≤ 𝑖 ≤ 𝑛 

f ( u′i ) = 6(i-1) and f (  u′′i ) = 6i - 2: 1≤ 𝑖 ≤ 𝑛 

It is evident that the included edge labels are both odd and distinct. 

Hence G = Pn ʘ K1,2 is an even vertex odd edge root square mean labelling. 

Illustration: Figure 3 illustrate the G = P4 ʘ K1,2 is even vertex odd edge root square mean labeling 

 

  

 

 

    

  

  

 
 

 

  

  



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Figure 3 G = P4 ʘ K1,2 

Theorem 3.4: The graph SLn  is an even vertex odd edge mean square labelling graph. 

Proof: Let G = SLn 

Let the vertex set V(G) = {𝑢𝑖, 𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛} and edge set  

E (G) = {

ui  ui + 1 ∶  1 ≤  i ≤  n − 1

vi  vi + 1 ∶  1 ≤  i ≤  n − 1 
vi+2  vi ∶  2 ≤  i ≤  n − 1

 

Define a map f: V(G) → {0,1, 2, ……2q} by f (u1) = 0, f(𝑣𝑛) = 2𝑞, further 

f (ui) = 6𝑖 – 10 ∶ 2 ≤ 𝑖 ≤ 𝑛, f (vi) = 6𝑖 – 2 ∶ 1 ≤ 𝑖 ≤ 𝑛 − 1 

Then the induced edge labels are defined as follows; 

𝑓′ (vivi+1) = 6𝑖 + 1∶ 1 ≤ 𝑖 ≤ 𝑛 − 2, 

𝑓′ (vn−1vn) = 2𝑞 – 1 ∶ 1 ≤ 𝑖 ≤ 𝑛 − 2 and 

𝑓′ (ui+1vi) = 6𝑖 − 3∶ 1 ≤ 𝑖 ≤ 𝑛 − 1. 

Hence,  SLn admits even vertex odd edge mean square labelling graph. 

Illustration: Figure 4 illustrate the G = SL6 is even vertex odd edge root square mean labeling 

 
Figure 4 G = SL6  

Theorem 3.5:[5] Let Gp be a graph formed by attaching a pendant edge to each internal vertex of a 

path. This graph Gp is an even vertex odd edge root mean square labelling graph. 

 Proof: 

 Let V(G) = { ui : 1 ≤  i ≤  n and u′i : 2 ≤  i ≤  n − 1   and  

 

 

 

 

 

 

 

 

 

 

  

  

  

  

  

  

  

  

  

  

  

  

  

         

           

        

    
  

  

         



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E(G) = {
ui ui+1  : 1 ≤  i ≤  n −  2   

ui u′i  ∶  i = 2,3,4,5 . . . . n − 2
 

be the vertex set and edge set of G respectively.  

Define a map f: V(G) →  { 0,2,4, . . . 2q}  by f(u) where f(u) is determined based on the following two 

cases. 

Case 1:  For n is odd  

f (ui) = 4(i -1) and f (u′i ) = 4i-6: i = 3, 5, …  

f (ui) = 4i -6 and f (u′i ) = 4(i-1): i = 2, 4, …  

The labels of the edges are then defined as follows 

f′(ui u′i ) = 4i – 5, i = 2, 4,…. n-1 and f′(ui ui+1 ) = 4i – 3, i = 1, 2 ,….n-1 

Case 2:  

For n is even  

f (ui) = 4i -5 and  

f (u′i ) = 4i-6: i = 3, 5, …  

f (ui) = 4i -6 and f (u′i ) = 4(i-1): i = 2, 4, …  

Then the induced edge labels are defined as follows: 

f′(𝐮𝐢 𝐮′𝐢 ) = 4i – 5, i = 2, 4,….n-1 and f′(𝐮𝐢 𝐮𝐢+𝟏 ) = 4i – 3, i = 1, 2 ,….n-1 

Hence f is an even vertex odd edge root square mean labelling of graph G. Therefore, G is classified 

as an even vertex odd edge root square mean labelling of graph with respect to f. 

Theorem 3.6: TW(Pn) is an even vertex odd edge root square mean labelling of graph. 

Proof: Let G = TW(Pn) 

                          V(G) = {
ui : 1 ≤  i ≤  n         and      
u′i  and u′′i  ∶ 2 ≤  i ≤  n  

 

  E(G) = {
uiui+1 : 1 ≤  i ≤  n − 1         and      

uiu′i  and uiu′′
i 

 ∶ 2 ≤  i ≤  n     

be the vertex set and edge set of G respectively. We define a map f: V(G) → 0,2,4, … 2q  by f(u) = 0;  

             f (u′i) = 2(i + j -1) i = 2 and j = 1, 2; f (u′i) = 6(i-1), i = 3,4, …n; f (u′′i) = 6i-10,  

i = 3, 4,…n 

Then the induced edge labels are defined as follows; 

f′(ui u′i ) = 6i – 9, f′(ui u′′i ) = 6i – 7, i = 2,4, …, n-1 and f′(ui ui+1 ) = 6i-5; i =1, 2,…n-1 

Hence G is an even vertex odd edge root square mean labelling of graph since f is an even 

vertex odd edge root square mean labelling of graph G. 



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Illustration: Figure 5 illustrate the Gp on 6 vertices is even vertex odd edge root square mean 

labeling 

 

Figure 5 Gp on 6 vertices 

Theorem 3.5:  K1,n   an even vertex odd edge mean square labelling graph. if n ≤ 3. 

Proof:      

Let the vertex and edge sets of K1,n be  defined as {u, v1, v2, … , vn}and {e1, e2, … , en}    respectively. 

For n ≤ 3 , it exhibits   an odd vertex even edge mean square labelling graph. This demonstrates 

the figures that are admitted as an odd vertex even edge mean square labelling graph. 

 

If n = 4 

K 1 ,4  is not even vertex and odd edge labelling graph 

 
Since the graph depicts that the edge label 5 repeated two times 

If n = 5 

         

       

    

 
 

    

  

 

 
 

 
 

 
  

 

 
     



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Since the graph depicts that the edge label 5 repeated two times 

In general, n > 3 

Case (i) 

Let f(u) = 0, then f (ui) = {2, 4, 6, …, 2q} 

In this case, (0,6), (0, 8), will get same labels, which is contradiction to the definition. 

Case (ii) 

Let f(u) = 2q, then no edge can get the label 1. 

Since (0, 2q) = √2(q)2 = √2 q, which is not possible. 

Case (iii) 

If f(u) = {2 or 4 or 6 or …. 2(q-1)}, then we get the repetition of odd edge labelling. 

So, clearly one can check labelling of a graph. 

Theorem 3.6 A subdivision of Pn ʘ K1 is an even vertex odd edge root square mean labelling 

graph. 

Proof:      

The vertex set and edge set of,  

G, is defined as follows; 

V(G) = {ui 
   ; 1≤ 𝑖 ≤ 𝑛} ∪  {vi 1 ≤  i ≤  n − 1}  ∪   {ui , u′′i  1 ≤  i ≤  n}   and 

E(G) ={  {ui u′i}  ∪ {ui , u′′i }  ∪ {ui vi}  ∪  {ui vi + 1 }   : 1≤ 𝑖 ≤ 𝑛-1 } 

We define a map f: V(G) → {0,1, 2, 3,…2q}. 

The labelling pattern of vertices can be defined as follows; 

f (ui)  = 4i ; i = 1,2   f (ui)  = 8i – 6 ; i = 3,4,5…, n-1 

f(vi)  = 6i ; i = 1,2   f (vi)  = 8i – 2 ; i = 3,4,5…, n 

f (u′i) = 8i -6 ; i = 1,2   f(u′i) = 8(i – 8) ; i = 3,4,5…, n-1 

f (u′′i) = 14(i-1); i = 1,2  f (u′′i) = 8i - 4; i = 3,4,5…, n-1 and f (un )  =  2(q − 1)  

 

   
 

  

 
 

  

 

      



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Then the induced edge labels are 

f ′ (uivi)  = 6i – 1; i = 1,2  f ′ (uivi)  = 8i – 3; i = 3,4,5…, n-1 

f ′(viui+1)  = 8i – 1; 1≤ 𝑖 ≤ 𝑛-1 

f ′ ( uiu′i) = 6i – 3; i = 1,2  f ′ ( uiu′i) = 8i – 7; i = 3,4,5…, n-1 

f ′ ( u′iu′′i) = 12i – 11; i = 1,2  f ′ ( u′iu′′i) = 8i – 5; i = 3,4,5…, n-1 

Thus, subdivision of G = Pn ʘ K1 admit even vertex odd edge root square mean labelling graph. 

Illustration: Figure 6 illustrate a subdivision of Pn ʘ K1 on 8 vertices is an even vertex odd edge root 

square mean labeling 

 

Figure 6 Pn ʘ K1 on 8 vertices 

Conclusion : 

In conclusion, this paper presents a new labeling pattern based on the root square mean, which has been 

successfully applied to path-related graphs, the subdivision of path corona graphs, and star graphs. The results 

demonstrate the effectiveness of this approach in enhancing graph labeling techniques and offer potential for 

further research in related areas. 

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