A note on edge irregularity strength of Dandelion graph
H. M. Nagesh · arXiv (Cornell University) · 2024
For a simple graph $G$, a vertex labeling $ϕ:V(G) \rightarrow \{1, 2,\ldots,k\}$ is called $k$-labeling. The weight of an edge $xy$ in $G$, written $w_ϕ(xy)$, is the sum of the labels of end vertices $x$ and $y$, i.e., $w_ϕ(xy)=ϕ(x)+ϕ(y)$. 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 $f$, $w_ϕ(e) eq w_ϕ(f)$. 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 $Δ(G) \geq \lceil \frac{|E(G)|+1}{2} \rceil$; and determine the bounds when $Δ(G) < \lceil \frac{|E(G)|+1}{2} \rceil $.