Volume 65 | Issue 5 | Year 2019 | Article Id. IJMTT-V65I5P517 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I5P517

In this work we consider finite, undirected, simple graphs ๐บ = (๐,๐ธ) with ๐ vertices and ๐ edges. The neighbourhood of a vertex ๐ฃ โ ๐(๐บ) is the set ๐๐บ(๐ฃ) of all the vertices adjacent to ๐ฃ in ๐บ. For a set ๐ โ ๐(๐บ), the open neighbourhood๐๐บ(๐) is defined to be โช๐ฃโ๐ ๐๐บ(๐ฃ) and the closed neighbourhood ๐๐บ[๐] = ๐๐บ(๐) โช ๐. The degree of a vertex ๐ฃ โ ๐(๐บ) is the number of edges of ๐บ incident with ๐ฃ and is denoted by ๐๐๐๐บ(๐ฃ) or ๐๐๐(๐ฃ). The maximum and the minimum degrees of the vertices of ๐บ are respectively denoted by ฮ(๐บ) and ๐ฟ(๐บ). A vertex of a degree 0 in ๐บ is called an isolated vertex and a vertex of degree 1 is called a pendent vertex or an end vertex of ๐บ. A vertex of a graph ๐บ is said to be a vertex of full degree if it is adjacent to all other vertices in ๐บ. A graph ๐บ is said to be regular of degree ๐ if every vertex of ๐บ has degree ๐. Such graphs are called r-regular graphs. The Dutch windmill graph๐ท๐ (๐) , is the graph obtained by taking ๐ copies of the cycle graph ๐ถ๐ with a vertex in common. The Butterfly graph(also called the bowtie graph and the hourglass graph) is a planar undirected graph with 5 vertices and 6 edges. It can be constructed by joining 2 copies of the cycle graph ๐ถ3 with a common vertex. It is denoted by ๐ต๐ . The ladder graph๐ฟ๐ is a planar undirected graph with 2n vertices and ๐ + 2(๐ โ 1) edges. The Ladder graph obtained as the cartesian product of two graphs one of which has only one edge: ๐ฟ๐,1 = ๐๐ ร ๐1.

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S. Balamurugan , M. Anitha , P. Aristotle , C. Karnan, "A Note on Open Support of a Graph under
Addition II," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 65, no. 5, pp. 115-119, 2019. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V65I5P517