Untitled Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 901 https://internationalpubls.com SP Mean E-Cordial Labeling 1M. Aishwarya, 2*V. Maheswari and 3V. Balaji 1Research Scholar, Department of Mathematics, Vels Institute of Science Technology and Advanced Studies, Chennai, Tamil Nadu, India. Email: aishwaryamuthazhagu@gmail.com 2,*Research Supervisor, Professor, Department of Mathematics, Vels Institute of Science Technology and Advanced Studies, Chennai, Tamil Nadu, India. Corresponding author: 3Associate Professor, Department of Mathematics, Sacred Heart College, Tirupattur, Tamil Nadu, India. Email: pulibala70@gmail.com Article History: Received: 12-11-2024 Revised: 10-12-2024 Accepted: 16-01-2025 Abstract: Assigning an integer to a vertices or edges is called a vertex or edge labeling respectively. Suppose G is a simple graph. Consider the function f for the edge set 𝑓: 𝑅 β†’ {0, 1}. For each vertex t ∈ 𝑇, define f(t)=βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝑅(𝐺) (mod2). The function f is known as an E-cordial labeling (E- CL) of G if |π‘Ÿπ‘“(0) βˆ’ π‘Ÿπ‘“(1)| ≀ 1, and |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1 where π‘Ÿπ‘“(0), π‘Ÿπ‘“(1)and 𝑑𝑓(0), 𝑑𝑓(1) are the number of edges and vertices labeled with 0 and labeled by 1 respectively. A graph that admits E-CL is said to be E-cordial graphs (E-CG). Based on the above definition we propose a novel labeling known as SP Mean E-cordial labeling (E-CL). In this paper, we study SP Mean E-CL of several families of graphs such as complete bipartite graphs, complete graphs, wheels, etc. Keywords: Mean E-cordial labeling, Wheel, Complete graphs 1. Introduction In this research, a graph is defined as a simple graph G. For graph theory definition and results we refer to F. Harary [1,2]. Rosa [3] introduce a new idea called labeling the graph. Assigning an integer to a vertices or edges is called a vertex or edge labeling respectively. Several labeling introduced by various authors [4,5,6,7,8,9]. Here we discuss with edge labeling. If a mapping 𝑓: 𝑅(𝐺) β†’ {0,1,2 … … , π‘ž} exists, a graph G is said to have elegant edges such that the induced mapping π‘“βˆ—: 𝑇(𝐺) β†’ {0, 1, 2, … . . , 𝑝 βˆ’ 1} given by 𝑓(𝑒) ≀ βˆ‘ 𝑓(𝑒𝑑)π‘šπ‘œπ‘‘|𝑑| , 𝑒𝑑 βˆˆπ‘…(𝐺). [10,11,12,13,14,15,16]. The label of vertex v under f is said to be f(v). A mapping 𝑓: 𝑇(𝐺) β†’{0, 1} is known as binary vertex labeling of G. Cordial labeling is the binary vertex labeling of a graph G if |π‘Ÿπ‘“(0) βˆ’ π‘Ÿπ‘“(1)| ≀ 1 and |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1. A graph satisfies cordial labeling is defined as cordial graph, where π‘Ÿπ‘“(0), π‘Ÿπ‘“(1) and 𝑑𝑓(0), 𝑑𝑓(1)are the number of edges and vertices labeled by 0 and 1 respectively. The cordial labeling introduced in 1987 by Cahit [17]. 2. SP Mean E-Cordial Labeling Definition 2.1 Let G be a simple graph. Let f be a function 𝑓:𝑅 β†’ {1,2} and an induced function π‘“βˆ—: 𝑇 β†’ {0,1}. We associate two integers. 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝑅(𝐺) and 𝑃 = ∏ 𝑓(𝑒𝑑). For each maheswari.sbs@vistas.ac.in mailto:aishwaryamuthazhagu@gmail.com mailto:sasikala.sbs@velsuniv.ac.in mailto:pulibala70@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 902 https://internationalpubls.com vertex u assign the label βŒŠπ‘†+𝑃2 βŒ‹ (mod 2), then f is defined as SP mean E-cordial labeling if |π‘Ÿπ‘“(1) βˆ’π‘Ÿπ‘“(2)| ≀ 1 and |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1 where 𝑑𝑓(0) and 𝑑𝑓(1) is the number of vertices labeled with 0 and labeled by 1 and π‘Ÿπ‘“(1) and π‘Ÿπ‘“(2) is the number of edges labeled with 1 and labeled by 2 respectively. A graph which admits SP mean E-CL is known as SP Mean E-CG (Figure 1). Example 2.2 Figure 1. SP Mean E-CG For vertex 𝑒1 , then 𝑆 = 1 and 𝑃 = 1 then π‘“βˆ—(𝑒1) = 1 For vertex 𝑒2 , then 𝑆 = 4 and 𝑃 = 2 then π‘“βˆ—(𝑒2) = 1 For vertex 𝑒3 , then 𝑆 = 4 and 𝑃 = 2 then π‘“βˆ—(𝑒3) = 0 For vertex 𝑒4 , then 𝑆 = 4 and 𝑃 = 2 then π‘“βˆ—(𝑒4) = 1 For vertex 𝑒5 , then 𝑆 = 5 and 𝑃 = 4 then π‘“βˆ—(𝑒5) = 0 For vertex 𝑒6 , then 𝑆 = 2 and 𝑃 = 2 then π‘“βˆ—(𝑒6) = 0 For vertex 𝑒7 , then 𝑆 = 2 and 𝑃 = 2 then π‘“βˆ—(𝑒7) = 0 Hence 𝑑𝑓(0) = 4, 𝑑𝑓(1) = 3 this implies |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1 Hence the above graph is a SP Mean E-CG. Theorem 2.3 For any Star (𝐾1,𝑛) is SP Mean E-cordial graph if n even. Proof: Let 𝑇(𝐾1,𝑛) = {𝑒, 𝑒𝑖 / 𝑖 = 1,2, … . 𝑛} be the vertices and 𝑅(𝐾1,𝑛) = {𝑒𝑒𝑖 / 𝑖 = 1,2, … . 𝑛} be the edges. Then | T(𝐾1,𝑛)| = 𝑛 + 1 and | E(𝐾1,𝑛)| = 𝑛 Define the function 𝑓: 𝑅(𝐺) β†’ {1,2} as 𝑓(𝑒𝑒𝑖) = {2 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛1 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ Find 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝐸(𝐺) and 𝑃 = ∏ 𝑓(𝑒𝑑) for each vertex in T(𝐾1,𝑛) Define π‘“βˆ—: 𝑇(𝐺) β†’ {0,1} defined by π‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹(mod 2). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 903 https://internationalpubls.com Then for 𝑛 ≑ 0(π‘šπ‘œπ‘‘ 4) 𝑑𝑓(0) 𝑑𝑓(1) π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑛≑ 0(π‘šπ‘œπ‘‘ 4) 𝑛2 𝑛2+ 1 𝑛2 𝑛2 𝑛≑ 0(π‘šπ‘œπ‘‘ 4) 𝑛2+ 1 𝑛2 𝑛2 𝑛2 The above labeling satisfies, |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1 and |π‘Ÿπ‘“(1) βˆ’ π‘Ÿπ‘“(2)| ≀ 1. The Star graph admits SP Mean E-CL. Hence star graph is a SP Mean E-CG if n is even. Theorem 2.4 For any Path graph 𝑃𝑛 is SP Mean E-CG if n is odd. Proof: Let 𝑇(𝑃𝑛) = {𝑒1, 𝑒2, … , 𝑒𝑛} and 𝑅(𝑃𝑛) = {𝑒1𝑒2, 𝑒2𝑒3, 𝑒3𝑒4, … , π‘’π‘›βˆ’1𝑒𝑛} be the vertices and edges respectively. Then | T(𝑃𝑛)| = 𝑛 and | R(𝐾1,𝑛)| = 𝑛 βˆ’ 1 Define the labeling function 𝑓: 𝑅(𝐺) β†’ {1,2} as follows 𝑓(𝑒𝑖𝑒𝑖+1) = { 1 𝑖𝑓 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 122 𝑖𝑓 𝑛 βˆ’ 12 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 Find 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝑅(𝐺) and 𝑃 = ∏ 𝑓(𝑒𝑑) for each vertex in 𝑇(𝐾1,𝑛) Defineπ‘“βˆ—: 𝑇(𝐺) β†’ {0,1} defined by π‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹ (mod 2). 𝑑𝑓(0) 𝑑𝑓(1) π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑛 + 12 𝑛2 𝑛 βˆ’ 12 𝑛 βˆ’ 12 The above labeling satisfies, and |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1 and |π‘Ÿπ‘“(1) βˆ’ π‘Ÿπ‘“(2)| ≀ 1. The Path graph admits SP Mean E-CL. Hence Path graph is a SP Mean E-CG. Theorem 2.5 For any Cycle graph 𝐢𝑛 is SP mean E-CG if n is odd. Proof: Let 𝑇(𝐢𝑛) = {𝑒1, 𝑒2, … , 𝑒𝑛} and 𝑅(𝐢𝑛) = {𝑒1𝑒2, 𝑒2𝑒3, 𝑒3𝑒4, … , π‘’π‘›βˆ’1𝑒𝑛, 𝑒𝑛𝑒𝑛1} be the vertices and edges respectively . Then | T(𝑃𝑛)| = 𝑛 and | R(𝐾1,𝑛)| = 𝑛 βˆ’ 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 904 https://internationalpubls.com Define the labeling function 𝑓: 𝑅(𝐺) β†’ {1,2} as follows 𝑓(𝑒𝑖𝑒𝑖+1) = { 1 𝑖𝑓 1 ≀ 𝑖 ≀ 𝑛 + 122 𝑖𝑓 𝑛 + 12 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 Find 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝑅(𝐺) and 𝑃 = ∏ 𝑓(𝑒𝑑) for each vertex in 𝑇(𝐾1,𝑛) Defineπ‘“βˆ—: 𝑇(𝐺) β†’ {0,1} defined byπ‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹ (mod 2). 𝑑𝑓(0) 𝑑𝑓(1) π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑛 + 12 𝑛 βˆ’ 12 𝑛 + 12 𝑛 βˆ’ 12 The above labeling satisfies,|𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1 and |π‘Ÿπ‘“(1) βˆ’ π‘Ÿπ‘“(2)| ≀ 1. The Cycle graph admits SP Mean E-CL. Hence Cycle graph is a SP Mean E-cordial graph. Note 2.6 𝐾3, 𝐾4 are the only complete graph in SP Mean E-cordial graph. Theorem 2.7 For any Wheel graph π‘Šπ‘›is a SP Mean E-cordial graph if n is even. Proof: Let 𝑇(π‘Šπ‘›) = {𝑒, 𝑒𝑖 𝑖 = 1,2,3 … 𝑛} and 𝑅(π‘Šπ‘›) = {𝑒𝑒𝑖 , 𝑒1𝑒2, 𝑒2𝑒3, 𝑒3𝑒4, … , 𝑒𝑛𝑒1} be the vertices and edges respectively, u is apex vertex and 𝑒1, 𝑒2, 𝑒3 … , 𝑒𝑛 be the vertices of cycle 𝐢𝑛. Then | T(π‘Šπ‘›)| = 𝑛 + 1 and | R(π‘Šπ‘›)| = 𝑛 Define the labeling 𝑓: 𝑅(𝐺) β†’ {1,2} as below 𝑓(𝑒𝑒𝑖) = 2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑓(𝑒𝑒𝑖) = 1, 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ 𝑓(𝑒𝑖𝑒𝑖+1) = 2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑓(𝑒𝑖𝑒𝑖+1) = 1, 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ Find 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝐸(𝐺) and 𝑃 = ∏ 𝑓(𝑒𝑑) for each vertex in 𝑇(𝐾1,𝑛) Define π‘“βˆ—: 𝑇(𝐺) β†’ {0,1} defined by π‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹ (mod 2). 𝑑𝑓(0) 𝑑𝑓(1) π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑛2 𝑛2 𝑛 𝑛 The above labeling satisfies |π‘Ÿπ‘“(1) βˆ’ π‘Ÿπ‘“(2)| ≀ 1 and |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 905 https://internationalpubls.com Hence Wheel graph π‘Šπ‘› admits SP Mean E-cordial graph labeling. Theorem 2.8 For any Bistar 𝐡𝑛,𝑛 is SP Mean E-CG if n is even. Proof: Let 𝑇(𝐡𝑛,𝑛) = {𝑒, 𝑑, 𝑒1, 𝑒2, 𝑒3 … . . 𝑒𝑛, 𝑑1, 𝑑2, 𝑑3 … . , 𝑑𝑛} be the vertices and 𝑅(𝐡𝑛,𝑛) = {𝑒𝑑, 𝑒𝑒1, 𝑒𝑒2, 𝑒𝑒3 … . . 𝑒𝑒𝑛, 𝑑𝑑1, 𝑑𝑑2, 𝑑𝑑3 … . , 𝑑𝑑𝑛} be the edges. Then | T(𝐡𝑛,𝑛)| = 2𝑛 + 2 and | R(π‘Šπ‘›)| = 2𝑛 + 1 To define the labeling function 𝑓: 𝑅(𝐺) β†’ {1,2} as follows 𝑓(𝑒𝑑) = 1 𝑓(𝑒𝑑𝑖) = 1 𝑓(𝑑𝑑𝑖) = 2 Find 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝑅(𝐺) and 𝑃 = ∏ 𝑓(𝑒𝑑) for each vertex in 𝑇(𝐾1,𝑛) Defineπ‘“βˆ—: 𝑇(𝐺) β†’ {0,1} defined by π‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹ (mod 2). 𝑑𝑓(0) 𝑑𝑓(1) π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑛 + 1 𝑛 + 1 𝑛 𝑛 + 1 The above labeling satisfies |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1 and |π‘Ÿπ‘“(𝑖) βˆ’ π‘Ÿπ‘“(𝑗)| ≀ 1. The Bistar 𝐡𝑛,𝑛admits SP Mean E-CL. Therefore, Bistar 𝐡𝑛,𝑛is a SP Mean E-CG. Theorem 2.9 Combo graph 𝑃𝑛 ⊚ 𝐾2 is a SP Mean E-CG. Proof: Let 𝑇(𝑃𝑛 ⊚ 𝐾2) = {𝑒𝑖 , 𝑒𝑖′1 ≀ 𝑖 ≀ 𝑛} be the vertices and 𝑅(𝑃𝑛 ⊚ 𝐾2) = {𝑒𝑖𝑒𝑖+1, 𝑒𝑒𝑖′, 1 ≀ 𝑖 ≀ 𝑛} be the edges. Then | T(𝑃𝑛 ⊚ 𝐾2)| = 2𝑛 and | R(𝑃𝑛 ⊚ 𝐾2)| = 2𝑛 βˆ’ 1 Define the function 𝑓: 𝑅(𝐺) β†’ {1,2} as below 𝑓(𝑒𝑖𝑒𝑖+1) = { 2 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛1 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ 𝑓(𝑒𝑖𝑒𝑖′) = {2 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛1 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ Find 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝐸(𝐺) and 𝑃 = ∏ 𝑓(𝑒𝑑) for each vertex in 𝑇(𝐾1,𝑛) Define π‘“βˆ—: 𝑇(𝐺) β†’ {0,1} defined by π‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹ (mod 2). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 906 https://internationalpubls.com π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑑𝑓(0) 𝑑𝑓(1) 𝑛 π‘›βˆ’ 1 𝑛 𝑛 The above labeling satisfies |π‘Ÿπ‘“(1) βˆ’ π‘Ÿπ‘“(2)| ≀ 1 and |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1. Combo graph 𝑃𝑛 ⊚ 𝐾2 admits SP Mean E-cordial labeling. Hence Combo graph 𝑃𝑛 ⊚ 𝐾2is a SP Mean E-cordial graph. Theorem 2.10 Crown graph 𝐢𝑛 ⊚ 𝐾2 is a SP Mean E-CG. Proof: Let 𝑇(𝐢𝑛 ⊚ 𝐾2) = {𝑒1, 𝑒2, 𝑒3 … , 𝑒𝑛, 𝑒1, , 𝑒2, , 𝑒3, … , 𝑒𝑛, } be the vertices and 𝑅(𝐢𝑛 ⊚ 𝐾2) ={𝑒1𝑒2, 𝑒2𝑒3, 𝑒3𝑒4, … , 𝑒𝑛𝑒1, 𝑒1𝑒1, , 𝑒2𝑒2, , 𝑒3𝑒3, … , 𝑒𝑛𝑒𝑛, } be the edges. Then | T(𝐢𝑛 ⊚ 𝐾2)| = 2𝑛 and | R(𝐢𝑛 ⊚ 𝐾2)| = 2𝑛 Define the labeling function 𝑓: 𝑅(𝐺) β†’ {1,2} as below For 1≀ 𝑖 ≀ 𝑛 βˆ’ 1 If n is odd 𝑓(𝑒𝑖𝑒𝑖+1) = {2 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛1 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ 𝑓(𝑒𝑖𝑒𝑖′) = {1 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛2 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ If n is even 𝑓(𝑒𝑖𝑒𝑖+1) = {2 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛1 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ 𝑓(𝑒𝑖𝑒𝑖′) = {2 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛1 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ Find 𝑃 = ∏ 𝑓(𝑒𝑑) and 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝐸(𝐺) for each vertex in 𝑇(𝐾1,𝑛) Defineπ‘“βˆ—: 𝑇(𝐺) β†’ {0,1} by π‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹ (mod 2). π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑑𝑓(0) 𝑑𝑓(1) 𝑛 is even 𝑛 𝑛 𝑛 𝑛 𝑛 is odd 𝑛 𝑛 βˆ’ 1 𝑛 𝑛 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 907 https://internationalpubls.com The above labeling satisfies |π‘Ÿπ‘“(1) βˆ’ 𝑑𝑓(2)| ≀ 1 and |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1. The Crown graph 𝐢𝑛 ⊚ 𝐾2admits SP Mean E-CL. Hence Crown graph 𝐢𝑛 ⊚ 𝐾2is a SP Mean E-cordial graph. Theorem 2.11 Triangular snake 𝑛𝐢3 is a SP Mean E-CG. Proof: Let the path 𝑃𝑛 having the edges π‘Ÿ1, π‘Ÿ2, π‘Ÿ3 … … . . , π‘Ÿπ‘›βˆ’1 and vertices 𝑑1, 𝑑2, 𝑑3, … , 𝑑𝑛. To construct Triangular snake 𝑛𝐢3 from path 𝑃𝑛 join 𝑒𝑖 and 𝑒𝑖+1 to a new edge 𝑑𝑖 by edges 𝑒𝑖𝑑𝑖 and 𝑒𝑖+1𝑑𝑖, for i = 1,2,3… … .., 𝑛 βˆ’ 1 Then | T(𝑛𝐢3)| = 2𝑛 βˆ’ 1 and | R(𝑛𝐢3)| = 3𝑛 βˆ’ 3 Define the labeling function 𝑓: 𝑅(𝐺) β†’ {1,2} 𝑓(𝑒𝑖𝑒𝑖+1) = { 2 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 1 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ 𝑓(𝑒𝑖𝑒𝑖′) = 1 𝑓(𝑒𝑖+1𝑒𝑖′) = 2 Find 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝐸(𝐺) and 𝑃 = ∏ 𝑓(𝑒𝑣) for each vertex in 𝑇(𝐾1,𝑛) Defineπ‘“βˆ—: 𝑇(𝐺) β†’ {0,1} defined by π‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹ (mod 2). 𝑑𝑓(0) 𝑑𝑓(1) π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑛 is even 𝑛 𝑛 βˆ’ 1 𝑛 + 1 𝑛 + 1 𝑛 is odd 𝑛 𝑛 βˆ’ 1 𝑛 + 1 𝑛 + 1 The above labeling satisfies β€œ|π‘Ÿπ‘“(1) βˆ’ π‘Ÿπ‘“(2)| ≀ 1 and |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1. The Triangular snake 𝑛𝐢3 admits SP Mean E-cordial labeling Hence Triangular snake 𝑛𝐢3is a SP Mean E-cordial graph. Theorem 2.12 Fan graph 𝐹1,𝑛 is a SP mean E-CG if n is odd. Proof: Let 𝑇(𝐹1,𝑛) = {𝑒, 𝑒1, 𝑒2, 𝑒3 … , 𝑒𝑛, } be the vertices and 𝑅(𝐹1,𝑛) ={𝑒1𝑒2, 𝑒2𝑒3, 𝑒3𝑒4, … , π‘’π‘›βˆ’1𝑒𝑛, 𝑒𝑒1, 𝑒𝑒2𝑒𝑒3, … , 𝑒𝑒𝑛} be the edges. Here u is apex vertex and 𝑒1, 𝑒2, 𝑒3 … , 𝑒𝑛 be the vertices of path. Then| R(𝐹1,𝑛)| =2𝑛 βˆ’ 1 and | T(𝐹1,𝑛)| = 𝑛 + 1. Define the function 𝑓: 𝑅(𝐺) β†’ {1,2} as below 𝑓(𝑒𝑒𝑖) = {2 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛1 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 908 https://internationalpubls.com 𝑓(𝑒𝑖𝑒𝑖+1) = {1 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛2 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ Find 𝑃 = ∏ 𝑓(𝑒𝑑) and 𝑆 = βˆ‘ 𝑓(𝑒𝑑) /𝑒𝑑 ∈ 𝐸(𝐺) for each vertex in 𝑇(𝐾1,𝑛) Defineπ‘“βˆ—: 𝑇(𝐺) β†’ {0,1} by π‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹ (mod 2). 𝑑𝑓(0) 𝑑𝑓(1) π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑛 + 12 𝑛 + 12 𝑛 𝑛 βˆ’ 1 The above labeling satisfies |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1 and |π‘Ÿπ‘“(1) βˆ’ π‘Ÿπ‘“(2)| ≀ 1. The Fan graph 𝐹1,𝑛 admits SP Mean E-cordial labeling. Hence Fan graph 𝐹1,𝑛 is a SP Mean E-CG. Theorem 2.13 Semi point total graph of path 𝑇2(𝑃𝑛)is a SP Mean E-CG. Proof: Let the path 𝑃𝑛 having the edges π‘Ÿ1, π‘Ÿ2, π‘Ÿ3 … … . . , π‘Ÿπ‘›βˆ’1 and the vertices 𝑒1, 𝑒2, 𝑒3, … 𝑒𝑛. To construct Semi point total graph of path 𝑇2(𝑃𝑛)from path 𝑃𝑛 join 𝑒𝑖 and 𝑒𝑖+1 to a new edge 𝑑𝑖 by edges 𝑒𝑖𝑑𝑖 and 𝑒𝑖+1𝑑𝑖, for i = 1,2,3… … .., 𝑛 βˆ’ 1. Then | T(𝑇2(𝑃𝑛))| =2𝑛 βˆ’ 1 and | R(𝑇2(𝑃𝑛))| = 3𝑛 βˆ’ 3 Define the labeling function 𝑓: 𝑅(𝐺) β†’ {1,2} 𝑓(𝑒𝑖𝑒𝑖+1) = { 2 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛1 𝑖𝑓 𝑖 𝑖𝑠 π‘œπ‘‘π‘‘ 𝑓(𝑒𝑖𝑒𝑖′) = 1 𝑓(𝑒𝑖+1𝑒𝑖′) = 2 Find 𝑆 = βˆ‘ 𝑓(𝑒𝑑) βˆ– 𝑒𝑑 ∈ 𝐸(𝐺) and 𝑃 = ∏ 𝑓(𝑒𝑑) for each vertex in 𝑇(𝐾1,𝑛) Define π‘“βˆ—: 𝑇(𝐺) β†’ {0,1} defined by π‘“βˆ—(𝑒) = βŒŠπ‘†+𝑃2 βŒ‹ (mod 2). π‘Ÿπ‘“(1) π‘Ÿπ‘“(2) 𝑑𝑓(0) 𝑑𝑓(1) 𝑛 is even 𝑛 + 1 𝑛 + 1 𝑛 𝑛 βˆ’ 1 𝑛 is odd 𝑛 + 1 𝑛 + 1 𝑛 𝑛 βˆ’ 1 The above labeling satisfies |π‘Ÿπ‘“(1) βˆ’ π‘Ÿπ‘“(2)| ≀ 1 and |𝑑𝑓(0) βˆ’ 𝑑𝑓(1)| ≀ 1.” The Semi point total graph of path 𝑇2(𝑃𝑛)admits SP Mean E-CL. 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