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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2405.13252 |
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| _version_ | 1866911885553565696 |
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| author | Nagesh, H. M. |
| author_facet | Nagesh, H. M. |
| contents | 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 $f$, $w_ϕ(e) \neq w_ϕ(f)$. 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 $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13252 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on edge irregularity strength of Dandelion graph Nagesh, H. M. Combinatorics 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 $f$, $w_ϕ(e) \neq w_ϕ(f)$. 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 $. |
| title | A note on edge irregularity strength of Dandelion graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.13252 |