Cordial and Total Cordial Labeling of Corona Product of Paths and Second Order of Lemniscate Graphs

A simple graph is called cordial if it admits 0-1 labeling that satisfies certain conditions. The second order of lemniscate graph is a graph of two second order of circles that have one vertex in common. In this paper, we introduce some new results on cordial labeling, total cordial, and present necessary and sufficient conditions of cordial and total cordial for corona product of paths and second order of lemniscate graphs.


Introduction
Labelling methods are used for a wide range of applications in di erent subjects including coding theory, computer science, and communication networks. Graph labeling is an assignment of positive integers on vertices or edges or both of them which ful lled certain conditions. Hundreds of research studies have been working with di erent types of labeling graphs [1][2][3][4][5][6][7][8][9][10][11], and a reference for this purpose is the survey written by Gallian [7]. All graphs considered, in this theme, are nite, simple, and undirected. e original concept of cordial graphs is due to Cahit [2]. He proved the following: each tree is cordial; a complete graph K n is cordial if and only if n ≤ 3 and a complete bipartite graph K n,m is cordial for all positive integers n and m [3]. Let G (V, E) be a graph, and let f: V ⟶ 0, 1 { } be a labeling of its vertices, and let the induced edge labeling f * E ⟶ 0, 1 { } be given by f * (uv) (f(u) + f(v))(mod2), where e uv(∈ E) and u, v ∈ V. Let v 0 and v 1 be the numbers of vertices that are labeled by 0 and 1, respectively, and let e 0 and e 1 be the corresponding numbers of edges. Such a labeling is called cordial if both |v 0 − v 1 | ≤ 1 and |e 0 − e 1 | ≤ 1 hold. A graph is called cordial if it admits a cordial labeling. As an extension of the cordial labeling, we de ne a total cordial labeling of a graph G with vertex set and edge set as an cordial labeling such that number of vertices and edges labeled with 0 and the number of vertices and edges labeled with 1 di er by at most 1, i.e., | (v 0 + e 0 ) − (e 1 +v 1 )| ≤ 1. A graph with a total cordial labeling is called a total cordial graph. If the vertices of the graph are assigned values subject to certain conditions, it is known as graph labeling. Following three are the common features of any graph labeling problem: (1) a set of numbers from which vertex labels are assigned; (2) a rule that assigns a value to each edge; and (3) a condition that these values must satisfy.
A path with n vertices and n − 1 edges is denoted by P n , and a cycle with n vertices and n edges is denoted by C n [12]. e second power of a lemniscate graph is de ned as the union of two second power of cycles where both have a common vertex; it is denoted by L 2 n,m ≡ C 2 n ♯C 2 m [13]. Obviously, L 2 n,m has n + m − 1 vertices and 2n + 2m − 4 edges. e corona product G 1 ⊙ G 2 of two graphs G i (with n i vertices and m i edges), i 1, 2, is the graph obtained by taking one copy of G 1 and n 1 copies of G 2 and then joining the i th vertex of G 1 with an edge to every vertex in the i th copy of G 2 . It is easy to show that G 1 ⊙ G 2 has n 1 (1 + n 2 ) vertices and m 1 + n 1 m 2 + n 1 n 2 edges [7,[14][15][16][17][18]. In this paper, we study the cordial and total cordial of the corona product P k ⊙ L 2 n,m of paths and second power of lemniscate graphs and show that this is cordial and total cordial for all positive integers k, n, m. e rest of the paper is organized as follows. In Section 1, brief summary of definitions that are useful for the present investigations is presented. Terminologies and notations are introduced in Section 2. e main result is presented in Section 3. Finally, the conclusion of this paper is introduced.

Results and Discussion
In this section, we show that the corona product of paths and second power of lemniscate graphs, P k ⊙ L 2 n,m , is cordial and also total cordial for all k ≥ 1, n, m ≥ 3. roughout our proofs, the way of labeling L 2 n,m starts always from a vertex that next the common vertex and go further opposite to this common vertex. Before considering the general form of the final result, let us first prove it in the following specific case. Our main theorem is as follows.

Theorem 1.
e corona product of paths and second power of lemniscate graphs, P k ⊙ L 2 n,m , is cordial and also total cordial for all k ≥ 1, n, m ≥ 3.
In order to prove this theorem, we will introduce a number of lemmas as follows.
3,m is cordial and total cordial for all k ≥ 1 and m ≥ 3.
3,4t+3 is cordial and total cordial. □ Lemma 2. P k ⊙ L 2 n,m is cordial and total cordial for all k ≥ 1 and m > 6.
Proof. Let k � 4r + i ' (i ' � 0, 1, 2, 3 and r ≥ 1) or k � 2r + j ' (j ' � 0, 1 and r ≥ 1), n � 4s + i and m � 4t + j (i, j � 1, 2, 3 and s, t ≥ 2), then we may use the labeling A i ' or A j ' for P k as given in Table 1. For a given value of j with 1 ≤ i, j ≤ 3, we may use one of the labeling in the set {B ij , ij are the labeling of L 2 n,m which are connected to the vertices labeled 0 in P k , while B ij and B ' ij are the labeling of P m which are connected to the vertices labeled 1 in P k as given in Table 2. Using Table 3

and the formulas
, we can compute the values shown in the last two columns of Table 3. We see that P k ⊙ L 2 n.m is isomorphic to P k ⊙ L 2 m,n . Since all of these values are 1 or 0, the lemma follows. □ Lemma 3. P k ⊙ L 2 4,m is cordial and total cordial for all k ≥ 1 and m > 3.
us, P 4r+3 ⊙ L 2 6,4t+3 is cordial and total cordial; by this, the lemma was proved, and through the proofs of these lemmas, we have completed the proof of our main theorem.

Conclusions
In this paper, we test the cordial and total cordial labeling of corona product of paths and second power of lemniscate graphs. We found that P k ⊙ L 2 n,m is cordial and also total cordial for all k ≥ 1, n, m ≥ 3. In future work, we can improve this work by using the different graphs with other mathematical operations to prove the cordial and total cordial labeling.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest. 8 Journal of Mathematics