A note on edge irregularity strength of Dandelion graph

TL;DR

This paper determines the exact edge irregularity strength of Dandelion graphs under certain maximum degree conditions and provides bounds in other cases, contributing to graph labeling theory.

Contribution

It finds the exact edge irregularity strength of Dandelion graphs for specific maximum degree conditions and establishes bounds otherwise.

Findings

Exact value of edge irregularity strength when Δ(G) ≥ ⌈(|E(G)|+1)/2⌉

Bounds on edge irregularity strength when Δ(G) < ⌈(|E(G)|+1)/2⌉

Enhanced understanding of edge irregular labelings in Dandelion graphs

Abstract

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 $.