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On the links of vertices in simplicial d-complexes embeddable in the euclidean 2d-space
, Article Discrete and Computational Geometry ; 2017 , Pages 1-17 ; 01795376 (ISSN) ; Sharif University of Technology
Abstract
We consider d-dimensional simplicial complexes which can be PL embedded in the 2d-dimensional Euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is linklessly embeddable in the (Formula presented.)-dimensional Euclidean space. In addition, we use similar considerations on links of vertices to derive a new asymptotic upper bound on the total number of d-simplices in an (continuously) embeddable complex in 2d-space with n vertices, improving known upper bounds, for all (Formula presented.). Moreover, we show that the same asymptotic bound also applies to the size of d-complexes linklessly embeddable in the...
On the links of vertices in eimplicial d-complexes embeddable in the euclidean 2d-space
, Article Discrete and Computational Geometry ; Volume 59, Issue 3 , 2018 , Pages 663-679 ; 01795376 (ISSN) ; Sharif University of Technology
Springer New York LLC
2018
Abstract
We consider d-dimensional simplicial complexes which can be PL embedded in the 2d-dimensional Euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is linklessly embeddable in the (2 d- 1 ) -dimensional Euclidean space. In addition, we use similar considerations on links of vertices to derive a new asymptotic upper bound on the total number of d-simplices in an (continuously) embeddable complex in 2d-space with n vertices, improving known upper bounds, for all d≥ 2. Moreover, we show that the same asymptotic bound also applies to the size of d-complexes linklessly embeddable in the (2 d+ 1 ) -dimensional...
Cohen–Macaulayness of two classes of circulant graphs
, Article Journal of Algebraic Combinatorics ; 2020 ; Maimani, H. R ; Mousivand, A ; Pournaki, M. R ; Sharif University of Technology
Springer
2020
Abstract
Let n be a positive integer and let Sn be the set of all nonnegative integers less than n which are relatively prime to n. In this paper, we discuss structural properties of circulant graphs generated by the Sn′s and their complements. In particular, we characterize when these graphs are well-covered, Cohen–Macaulay, Buchsbaum or Gorenstein. © 2020, Springer Science+Business Media, LLC, part of Springer Nature
Structural Properties of a Class of Cayley Graphs and their Complements: Well-coveredness and Cohen–Macaulayness
, M.Sc. Thesis Sharif University of Technology ; Pournaki, Mohammad Reza (Supervisor)
Abstract
Let be a field and R= [x0, . . . , xn−1] be the polynomial ring in n variables over the field . Let G be a finite simple graph with the vertex set V(G) ={0, . . . , n − 1} and the edge set E(G). One can associate a square-free quadratic monomial ideal I(G) = (xixj | {i, j} ∈ E(G)) of R to the graph G. The ideal I(G) is called the edge ideal of G in R. The graph G is called Cohen–Macaulay (resp. Buchsbaum, Gorenstein) over if the ring R/I(G) is Cohen–Macaulay (resp. Buchsbaum, Gorenstein).Let n be a positive integer and let Sn be the set of all nonnegative integers less than n which are relatively prime to n. In this thesis, we investigate structural properties of Cayley graphs generated by...
Cohen–Macaulayness of a Class of Graphs Due to Grimaldi
,
M.Sc. Thesis
Sharif University of Technology
;
Pournaki, Mohammad Reza
(Supervisor)
Abstract
Let K be a field and S=K[x0,…,xn-1] be the polynomial ring in n variables over the field K. Let G be a finite undirected graph without loops or multiple edges with the vertex set V(G)={0,…,n-1} and the edge set E(G). One can associate a squarefree quadratic monomial ideal I(G)=of S to the graph G. The ideal I (G) is called the edge ideal of G in S. It is an algebraic object whose invariants can be related to the properties of G and vice versa. The graph G is called Cohen–Macaulay over K (Gorenstein over K) if the ring S/I (G) is Cohen–Macaulay (Gorenstein). Let n ≥ 2 be an integer. The Grimaldi graph represented by G(n) is obtained by letting all the elements of...
A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra
, Article Research in Mathematical Sciences ; Volume 9, Issue 2 , 2022 ; 25220144 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Terai, N ; Yassemi, S ; Sharif University of Technology
Springer Science and Business Media Deutschland GmbH
2022
Abstract
A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG