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On the essential graph of a poset | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 08 تیر 1404 | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2025.29478.2093 | ||
| نویسندگان | ||
| Reza Nikandish1؛ Esmail Eskandari2؛ Amin Motamedinasab* 2 | ||
| 1Department of Mathematics, Jundi-Shapur University of Technology, P.O. BOX 64615-334, Dezful, Iran | ||
| 2Department of Physics, Technical and Vocational University (TVU), Tehran, Iran | ||
| چکیده | ||
| Let $(P, \leq)$ be an atomic partially ordered set (briefly, a poset) with a minimum element $0$, and let $\mathcal{I}(P)$ be the set of all nontrivial ideals of $ P $. The essential graph of $P$, denoted by $G_e(P)$, is an undirected, simple graph with the vertex set $\mathcal{I}(P)$ and two distinct vertices $I, J \in \mathcal{I}(P) $ are adjacent in $G_e(P)$ if and only if $ I\cup J $ is an essential ideal of $P$. We study the connections between the graph-theoretic properties of this graph and the algebraic properties of a poset. We prove that $G_e(P)$ is connected with diameter at most three. Furthermore, all posets are characterized based on the diameters of their essential graphs. Also, all posets with planar $G_e(P)$ are classified. Among other results, the clique number and chromatic number of $G_e(P)$ are determined. | ||
| کلیدواژهها | ||
| Essential graph؛ Diameter؛ Planar؛ Clique number | ||
| مراجع | ||
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[6] M.J. Nikmehr, R. Nikandish, and M. Bakhtyiari, On the essential graph of a commutative ring, J. Algebra Appl. 16 (2017), no. 7, Article ID: 1750132. https://doi.org/10.1142/S0219498817501328
[7] S.K. Nimbhorkar, M.P. Wasadikar, and L. DeMeyer, Coloring of semilattices, Ars Combin. 12 (2007), 97–104.
[8] S. Roman, Lattices and Ordered Sets, Springer Science & Business Media, New York, 2008.
[9] S. Rudeanu, Sets and Ordered Structures, Bentham Science Publishers, 2012.
[10] D.B. West, Introduction to Graph Theory, Prentice hall Upper Saddle River, 2001. | ||
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