On edge irregularity strength of cycle-star graphs
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| Format: | Preprint |
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2024
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| author | Salma, Umme Nagesh, H. M. N, Narahari |
| author_facet | Salma, Umme Nagesh, H. M. N, Narahari |
| 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 $uv$ in $G$, written $w_ϕ(uv)$, is the sum of the labels of end vertices $u$ and $v$, i.e., $w_ϕ(uv)=ϕ(u)+ϕ(v)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two distinct edges $u$ and $v$, $w_ϕ(u) \neq w_ϕ(v)$. 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 paper, we study the edge irregular $k$-labeling for cycle-star graph $CS_{k,n-k}$ and determine the exact value for cycle-star graph for $3 \leq k \leq 7$ and $n-k \geq 1$. Finally, we make a conjecture for the edge irregularity strength of $CS_{k,n-k}$ for $k \geq 8$ and $n-k \geq 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12263 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On edge irregularity strength of cycle-star graphs Salma, Umme Nagesh, H. M. N, Narahari 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 $uv$ in $G$, written $w_ϕ(uv)$, is the sum of the labels of end vertices $u$ and $v$, i.e., $w_ϕ(uv)=ϕ(u)+ϕ(v)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two distinct edges $u$ and $v$, $w_ϕ(u) \neq w_ϕ(v)$. 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 paper, we study the edge irregular $k$-labeling for cycle-star graph $CS_{k,n-k}$ and determine the exact value for cycle-star graph for $3 \leq k \leq 7$ and $n-k \geq 1$. Finally, we make a conjecture for the edge irregularity strength of $CS_{k,n-k}$ for $k \geq 8$ and $n-k \geq 1$. |
| title | On edge irregularity strength of cycle-star graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.12263 |