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.

No abstract

In this paper, some basic properties and applications of alpha dot-neighbourhoods, alpha dot-derived sets, alpha dot-interiorand alpha dot-closure are introduced and discussed by using alpha dot-closed sets in topological spaces.