Graph labelling is one of the major research area in Mathematics which was introduced by A. Rosa. A new concept of labelling namely Narayana Prime cordial labelling was introduced and studied by Murali, Thirusangu and Balamurugan. In this paper, we investigate the existence of Narayana prime cordial labelling of generalized Petersen graph.

The objective of this paper is to investigate some $k$-odd sequential harmonious labeling of graphs. In particular, we show that $k$-odd sequential harmonious labeling of double quadrilateral snakes $\left(2 Q_x\right.$-snakes) for each $x \geq 1$. We also prove that, $2 m Q_x$-snakes are $k$-odd sequential harmonious labeling of graphs for each $m, x \geq 1$. Finally, we present some examples and verified to illustrate proposed theories.

Intuitionistic Fuzzy set (IFS) was proposed in early 80’s. It is a well known theory. As a developer in Fuzzy Mathematics, interval – valued Intuitionistic Fuzzy sets (IVIFS) were developed afterwards by Gargo and Atanssov. It has a wide range of applications in the field of Optimization and algebra. There are many distance measure in Fuzzy such as Hamming, Normalized Hamming, Euclidean, Normalized Euclidean, Geometric, Normalized Geometric etc… to calculate the distance between two fuzzy numbers. In this paper, the comparison between Geometric distance measure in Intuitionistic Fuzzy set and interval – valued Intuitionistic Fuzzy sets is explored. The step-wise conservation of Intuitionistic Fuzzy set and interval – valued Intuitionistic Fuzzy sets is also proposed. This type of comparative analysis shows that the distance between Intuitionistic Fuzzy set and interval – valued Intuitionistic Fuzzy sets varies slightly due to boundaries of interval – valued Intuitionistic Fuzzy sets.

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In this paper we have introduced Bipolar Interval Valued Intuitionistic Fuzzy Point, Bipolar Interval Valued Intuitionistic Fuzzy Neighbourhood,Bipolar Interval Valued Intuitionistic Fuzzy Interior and Bipolar Interval Valued Intuitionistic Fuzzy Closurein Bipolar Interval Valued Intuitionistic Fuzzy topological space and have verifiedsome of its properties.