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A harmonious coloring of a simple graph G is a proper coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number ?_h (G) is the least number of colors in such a coloring. This parameter was first proposed by Frank O, Harary F and Plantholt M in 1982. However, the proper definition of this notion is due to Hopcroft J and Krishnamoorthy M.S in 1983. Every graph has a harmonious coloring, since it suffices to assign every vertex a distinct color; thus ?_h (G)?|V(G)|. It was shown by Hopcroft and Krishnamoorthy that the problem of determining the harmonious chromatic number of a graph is NP-hard. Zhikang Lu has obtained estimates for the harmonious chromatic numbers of some classes of graphs. He has also determined bounds on the harmonious chromatic number of a complete binary and trinary tree, and complete 4-ary tree. Later Lu provided the exact value of the harmonious chromatic number of a complete binary tree and trinary tree. At this point it is worth noticing that the problem is even harder when restricted to many graph families in which NP-hard problems usually become tractable. There are only a few families for which we can have exact solutions in polynomial time. In this paper we compute the harmonious coloring of central graph of Triangular graphs with chromatic number at most ?[C(G) ]+?(G). Triangular graphs like triangular snake graph, double triangular snake, alternate triangular snake, alternate double triangular snake, triple triangular snake and friendship graph are considered and results were determined. Finally we develop a common algorithm for harmonious coloring of central graph of graphs and the harmonious chromatic number of the graphs are also solved in polynomial time.

4.40 PM
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Digital topology plays a vital role in Digital image processing. The theory of nano topology was proposed by Lellis Thivagar, on a non-empty universe U in terms of lower, upper and boundary approximation of a subset of U with a equivalence relation. In 1979, digital topology was introduced by Azriel Rosenfeld. In computer vision, image segmentation is the way toward apportioning a picture into different fragments. K-Means clustering algorithm is utilized to recognize various classes or groups in the given information dependent on how do we compare the information. Data points in the same group are more similar to other data points in that same group than those in other groups. Also it is one of the maximum generally used clustering algorithms. Here, k represents the number of clusters. A definitive perspective on this paper is to tackle an Image segmentation problem by K Means clustering algorithm with aid of some form of open sets. For this, shape a brand new topology on a non empty set with widely known closure and interior operators and called digital nano topology. Further define digital nano topology on a digital plane Z ? Z with khalimsky plane topology, named as digital nano khalimsky plane (shortly, Digital NK) topology and some of its topological properties are studied. Using these novel concept, we deal real life application.

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In a simple connected graph G, we have defined and illustrated weakly connected point set dominating set(wcps), Dominating weakly connected point set dominating set(wcpsd) and the minimum cardinality of wcpsd set denoted wcpsd ? . Now we have described certain class of dominating weakly connected point set dominating sets of a graph. We have partitioned the vertices of a graph G into three sets according to how their removal affects (G) wcpsd ? . We have defined that( ) { / ( ) ( )} 0 Vwcpsd G ? v ?V ? wcpsd G ? v ? ? wcpsd G , V (G) {v V / (G v) (G)} wcpsd wcpsd wcpsd ? ? ? ? ? ??V (G) {v V / (G v) (G)} wcpsd wcpsd wcpsd ? ? ? ? ? ??such that ? ? V ? Vwcpsd ?Vwcpsd ?Vwcpsd0. We have observed that these class of sets may be empty for some graphs we described that necessary and sufficient condition for a vertex v to be in? Vwcpsd and to be in ? Vwcpsd . Further we obtained that the relation between the cardinality of these class of sets. Also we described some results on these class of sets.

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The theory of nano topology was introduced by Lellis Thivagar, on a non emptyuniverse U in terms of lower, upper and boundary approximation of a subset of U with a equivalence relation. Using nano topology many real life problems are solved. A multifunction is a function which takes each element of a domain into a set. Instead of indiscernibility relation,take a surjective multifunction to structure a new topology which contains at most five elements named as nano multifunctional (shortly, Nmf ) topology. Along with Nmf closure and Nmf interior are described and their properties had been studied. Together with Nmf extremally disconnected Nmf topological spaces in terms of Nmf closure of Nmf open is derived.

5.10 PM
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In this paper, we introduce a new class of open sets called semi b-open sets
in N-topological space. Also we explore the idea about semi b-connectedness in N-topology and obtain its rudimentary properties. In addition, we investigate semi b-connectedness and its relationship with some other mappings.

5.20 PM
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5.30 PM

Let D=(V,A) be a digraph. A subset S of arc in a digraph D is called an arc dominating set of D if for every , there exists an such that, The minimum cardinality of an arc dominating set of D is called the arc domination number of D and is donated by . A set of edges of D is twin arc dominating set if every edge of D is out arc dominated by some edge of S and in arc dominated by some edge of S. The minimum Cardinality of a twin arc dominating set is the twin arc domination number denoted by ?^* (D). A set S of arcs in a digraph is called an arc independent set if no two arcs in S are adjacent arcs. In this paper, arc independent set and twin arc domination for some standard digraphs were studied and some bounds between them were discussed.

5.30 PM to 5.40 PM

The concept of Triple connected domination number was introduced by G. Mahadevan et. al., The concept of eternal domination in graphs was introduced by W. Goddard., et. al., The dominating set S_0 (⊆V(G)) of the graph G is said to be an eternal dominating set, if for any sequence v_1,v_2,v_3,. . . v_k of vertices, there exists a sequence of vertices u_1,u_2,u_3,. . . u_k with u_1∈S_(i-1) and u_i equal to or adjacent to v_i, such that each set S_i=S_(i-1)-{u_i }∪{v_i} is dominating set in G. The minimum cardinality taken over the eternal dominating sets in G is called the eternal domination number of G and it is denoted by γ_∞ (G). The eternal dominating set S_0 (⊆V(G)) of the graph G is said to be a triple connected eternal dominating set, if each dominating set S_i is triple connected. The minimum cardinality taken over the triple connected eternal dominating sets in G is called the triple connected eternal domination number of G and it is denoted by γ_(tc,∞) (G). In this paper we investigate this number for different product of cycles.