Duplicating a Vertex with an Edge in Divided Square Difference Cordial Graphs

By a graph, we mean a finite, undirected graph without loops and multiple edges. For basic definitions we refer Harary [8]. In 1967, Rosa [10] introduced a labeling of G called β-valuation. A dynamic survey on different graph labeling was found in Gallian [7]. Cordial labeling was introduced by Cahit [5]. R.Varatharajan, et. al [11] have introduced the notion of divisor cordial labeling. A. Alfred Leo et.al [1] introduced divided square difference cordial labeling graphs. V.J. Kaneria et. al [9] introduced balanced cordial labeling. The motivation behind the divided square difference cordial labeling is due to R. Dhavaseelan et.al on their work even sum cordial labeling graphs [6]. The motivation behind this article is due to S.K. Vaidya et.al on their work [12]. In this present work, we discuss divided square difference (DSD) cordial labeling in the context of duplication of a vertex by an edge in DSD cordial graphs such as path graph, cycle graph, star graph, wheel graph, helm graph, bistar graph, crown graph, comb graph and snake graph.


Introduction
By a graph, we mean a finite, undirected graph without loops and multiple edges. For basic definitions we refer Harary [8]. In 1967, Rosa [10] introduced a labeling of G called β-valuation. A dynamic survey on different graph labeling was found in Gallian [7]. Cordial labeling was introduced by Cahit [5]. R. Varatharajan, et. al [11] have introduced the notion of divisor cordial labeling. A. Alfred Leo et.al [1] introduced divided square difference cordial labeling graphs. V.J. Kaneria et. al [9] introduced balanced cordial labeling. The motivation behind the divided square difference cordial labeling is due to R. Dhavaseelan et.al on their work even sum cordial labeling graphs [6]. The motivation behind this article is due to S.K. Vaidya et.al on their work [12]. In this present work, we discuss divided square difference (DSD) cordial labeling in the context of duplication of a vertex by an edge in DSD cordial graphs such as path graph, cycle graph, star graph, wheel graph, helm graph, bistar graph, crown graph, comb graph and snake graph.

Preliminaries
Definition 2.1 [7] Graph labeling is an assignment of numbers to the edges or vertices or both subject to certain condition(s). Proposition 2.8 [3] 1. The crown graph C K n  1 is DSD cordial. 2. The comb graph P K n  1 is DSD cordial.

Proposition 2.9 [4]
The triangular snake graph T n (except n ≡ 3 4 mod ) is a balanced DSD cordial when n is odd.

Proposition 3.1
A graph got by duplicating a vertex v k with an edge

Proof
Let G be a path graph P n (except n mod ≡ ( ) Hence ′ G is also a DSD cordial. Hence, we can conclude that ′ G is a balanced DSD cordial graph when n is even and unbalanced DSD cordial when n is odd.

Note 3.3
For n mod ≡ ( ) 2 4 , the path graph P n is DSD cordial whereas ′ G obtained by duplicating any of the vertex with an edge in P n is not DSD cordial.

Proposition 3.5
A graph got by duplicating a vertex v k with an edge . By Proposition 2.6, we draw a DSD cordial cycle graph C n . Now, we Hence ′ G is also a DSD cordial. Hence, we can conclude that ′ G is a balanced DSD cordial graph when n is odd and unbalanced DSD cordial graph when n is even.

Note 3.7
For n mod ≡ ( ) 3 4 , the cycle graph C n is DSD cordial whereas ′ G obtained by duplicating any of the vertex with an edge in C n is not DSD cordial.  Hence ′ G is also a DSD cordial. Hence, we can conclude that ′ G is a unbalanced DSD cordial.   Hence ′ G is also a DSD cordial.   Hence ′ G is also a DSD cordial.

Remark 3.16
From Proposition 3.15, in particular we get Hence, we can conclude that is a balanced DSD cordial graph when ′ G and unbalanced DSD cordial graph when n is even.

Proposition 3.18
A graph got by duplicating a vertex v k with an edge

Proof
Let G be a crown graph C K n  1 . By Proposition 2.8, we draw a DSD cordial crown graph C K n  1 . Now, Hence ′ G is also a DSD cordial.

Remark 3.19
From Proposition 3.18, in particular we get Hence, we can conclude that ′ G is a unbalanced DSD cordial graph.

Proposition 3.21
A graph got by duplicating a vertex v k with an edge ′ = ′ ′ e u v in a DSD cordial comb graph P K n  1 (except Hence ′ G is also a DSD cordial.

Remark 3.22
From Proposition 3.21, in particular we get Hence, we can conclude that ′ G is a balanced DSD cordial graph.

Note 3.23
For n mod ≡ ( ) 1 4 , the comb graph P K n  1 is DSD cordial whereas ′ G obtained by duplicating any of the vertex with an edge in P K n  1 is not DSD cordial.

Proposition 3.25
A graph got by duplicating a vertex v k with an edge ′ = ′ ′ e u v in a DSD cordial triangular snake graph T n (except n ≡ 2 4 mod ) is divided square difference cordial. pp.8

For n mod
≡ ( ) 3 4 , the triangular snake graph T n is not DSD cordial whereas ′ G got by duplicating any of the vertex with an edge in T n is DSD cordial.

Conclusion
In this article, we have discussed and proven that the graph got by duplicating a vertex with an edge in divided square difference (DSD) cordial graphs such as path graph, cycle graph, star graph, wheel graph, helm graph, bistar graph, crown graph, comb graph and snake graph were also DSD cordial graphs.