For a simple graph $G$, a vertex labeling $\phi:V(G) \rightarrow \{ 1, 2,\ldots,k\}$ is called $k$-labeling. The weight of an edge $xy$ in $G$, written $w_{ \phi}(xy)$, is the sum of the labels of end vertices $x$ and $y$, i.e., $w_{ \phi}(xy)=\phi(x)+\phi(y)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$, $w_{ \phi}(e) \neq w_{ \phi}(f)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, written $es(G)$. In this note, we find the exact value of edge irregularity strength of Dandelion graph when $\Delta(G) \geq \lceil \frac{ |E(G)|+1}{ 2} \rceil$; and determine the bounds when $\Delta(G) < \lceil \frac{ |E(G)|+1}{ 2} \rceil $.
Categories: math.CO