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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.02056 |
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| _version_ | 1866913340159164416 |
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| author | Jamir, Yangersenba T Dutta, S |
| author_facet | Jamir, Yangersenba T Dutta, S |
| contents | Let $C(X)$ be the ring of all continuous real valued functions defined on a completely regular Hausdorff topological space $X$. The zero-set intersection graph $Γ(C(X))$ of $C(X)$ is a simple graph with vertex set all non units of $C(X)$ and two vertices are adjacent if the intersection of the zero sets of the functions is non empty. In this paper, we study the zero-set intersection graph of $C(X)$ and its line graph. We show that if $X$ has more than two points, then these graphs are connected with diameter and radius 2. We show that the girth of the graph is 3 and the graphs are both triangulated and hypertriangulated. We find the domination number of these graphs and finally we prove that $C(X)$ is a von Neuman regular ring if and only if $C(X)$ is an almost regular ring and for all $f \in V(Γ(C(X)))$ there exists $g \in V(Γ(C(X)))$ such that $Z(f) \cap Z(g) = ϕ$ and $\{f, g\}$ dominates $Γ(C(X))$. Finally, we derive some properties of the line graph of $Γ(C(X))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some properties of the Zero-set Intersection graph of $C(X)$ and its Line graph Jamir, Yangersenba T Dutta, S Combinatorics 54C40, 05C69 Let $C(X)$ be the ring of all continuous real valued functions defined on a completely regular Hausdorff topological space $X$. The zero-set intersection graph $Γ(C(X))$ of $C(X)$ is a simple graph with vertex set all non units of $C(X)$ and two vertices are adjacent if the intersection of the zero sets of the functions is non empty. In this paper, we study the zero-set intersection graph of $C(X)$ and its line graph. We show that if $X$ has more than two points, then these graphs are connected with diameter and radius 2. We show that the girth of the graph is 3 and the graphs are both triangulated and hypertriangulated. We find the domination number of these graphs and finally we prove that $C(X)$ is a von Neuman regular ring if and only if $C(X)$ is an almost regular ring and for all $f \in V(Γ(C(X)))$ there exists $g \in V(Γ(C(X)))$ such that $Z(f) \cap Z(g) = ϕ$ and $\{f, g\}$ dominates $Γ(C(X))$. Finally, we derive some properties of the line graph of $Γ(C(X))$. |
| title | Some properties of the Zero-set Intersection graph of $C(X)$ and its Line graph |
| topic | Combinatorics 54C40, 05C69 |
| url | https://arxiv.org/abs/2405.02056 |