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Remarks on the zero-set intersection graph | ||
| Journal of Algebra and Related Topics | ||
| دوره 13، شماره 2، اسفند 2025، صفحه 99-108 اصل مقاله (162.89 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2025.27038.1652 | ||
| نویسندگان | ||
| P. Dhar* ؛ J. P. J. Kharbhih | ||
| Department of Mathematics, North-Eastern Hill University, Shillong, India | ||
| چکیده | ||
| In this paper, we study the zero-set intersection graph ($\Gamma(C(X))$) and its line graph ($L(\Gamma(C(X)))$). We showed that $0$ is a cut vertex of $\Gamma(C(X))$ iff $|X|=2$, and for a first countable space $X$, $\Gamma(C(X))$ is chordal iff $|X|=2 \ or\ |X|=3.$ We stated some conditions for a maximal clique to be a maximal ideal. We obtained that two (first countable/real compact) topological spaces $X$ and $Y$ are homeomorphic iff $L(\Gamma(C(X)))$ is graph isomorphic to $L(\Gamma(C(Y)))$ iff $C(X)$ is isomorphic to $C(Y).$ We showed that $\{f,g\}$ is a dominating set of $\Gamma(C(X))$ iff $fg=0.$ | ||
| کلیدواژهها | ||
| Line graph؛ Triangulated؛ Chordal؛ Complemented؛ Ring of real-valued continuous functions | ||
| مراجع | ||
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[1] G. AlAfifi and E.Abu Osba, On the line graph for zero-divisors of C(X), Hindawi Publishing Corporation, International journal of combinatorics, 2013 (2013), 1-6. [2] F. Azarpanah and M. Motamedi, Zero-divisor graph of C(X), Acta Math. Hungar., 108 (2005), 25-36. [3] B. Bose and A. Das, Graph theoretic representation of rings of continuous functions, Filomat, (10) 34 (2020), 3417-3428. [4] W. W. Comfort and A. W. Hager, Estimates for the number of real-valued continuous functions, Trans. Amer. Math. Soc., 150 (1970), 619-631. [5] R. Diestel, Graph Theory, 3rd edition, Graduate Texts in Mathematics, 173, Springer-Verlag, Berlin, 2005. [6] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1978. [7] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc., 115 (1965), 110-130. [8] M. Venkatachalapathy, K. Kokila and B. Abarna, Some trends in line graphs, Advances in Theoretical and Applied Mathematics, (2) 11 (2016), 171-178. | ||
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آمار تعداد مشاهده مقاله: 2 تعداد دریافت فایل اصل مقاله: 1 |
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