On a graph of ideals

Amini, Afshin; Amini, Babak; Momtahan, Ehsan; Haghighi, Mohammad Hassan Shirdareh

doi:10.1007/s10474-011-0121-3

Cited by 16 publications

(8 citation statements)

References 14 publications

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“…In these investigations were tried to associate the ring properties of C(X), the graph properties of graphs on C(X) and the topological properties of X. In [5,3,6] the zero-divisor graph, the comaximal ideal graph of C(X) and comaximal graph of C(X) were studied. In [7,8], the studying annihilating-ideal graph of commutative rings were started.…”

Section: Clearly Clique(g) χ(G)mentioning

confidence: 99%

Annihilating-Ideal Graph of $C(X)$

Badie

2018

Preprint

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In this article we study the annihilating-ideal graph of the ring C(X). We have tried to associate the graph properties of AG(X), the ring properties of C(X) and the topological properties of X. We have shown that X has an isolated point if and only if R is a direct summand of C(X) if and only if AG(X) is not triangulated. Radius, girth, dominating number and clique number of the AG(X) are investigated. We have proved that c(X) dt(AG(X)) w(X) and cliqueAG(X) = χAG(X) = c(X).

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Section: Clearly Clique(g) χ(G)mentioning

confidence: 99%

Annihilating-Ideal Graph of $C(X)$

Badie

2018

Preprint

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“…However, as their work was a follow up of zero divisor graph of a ring, the main topological flavor of characterizing the graph was missing. Apart from that, Amini et al [2] and Badie [22] also studied another graph structure on C(X) from a co-maximal ideal point of view.…”

Section: Introductionmentioning

confidence: 99%

Graph theoretic representation of rings of continuous functions

Bose

Das

2020

Filomat

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In this paper, we introduce a graph structure, called zero-set intersection graph ?(C(X)), on the ring of real valued continuous functions, C(X), on a Tychonoff space X. We show that the graph is connected and triangulated. We also study the inter-relationship of cliques of ?(C(X)) and ideals in C(X) which helps to characterize the structure of maximal cliques of ?(C(X)) by different kind of maximal ideals of C(X). We show that there are at least 2c many different maximal cliques which are never graph isomorphic to each other. Furthermore, we study the neighbourhood properties of a vertex and show its connection with the topology of X and algebraic properties of C(X). Finally, it is shown that two graphs are isomorphic if and only if the corresponding rings are isomorphic if and only if the corresponding topologies are homeomorphic either for first countable topological spaces or for realcompact topological spaces.

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“…The study of graphs associated with of various algebraic structures was initiated by Beck [4] who introduced the idea of zero divisor graph of a commutative ring with unity. Till then, a lot of research, e.g., [13,2,3,1,8,6,7,12,16] has been done in connecting graph structures to various algebraic objects like semigroups, groups, rings, vector spaces etc. In this paper, we continue the study of one such graph called Non-Zero Component Graph [9] of a finite dimensional vector space V over a field F with respect to a basis {α 1 , α 2 , .…”

Section: Introductionmentioning

confidence: 99%

On nonzero component graph of vector spaces over finite fields

Das

2017

J. Algebra Appl.

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In this paper, we study nonzero component graph Γ(V) of a finite-dimensional vector space V over a finite field F. We show that the graph is Hamiltonian and not Eulerian. We also characterize the maximal cliques in Γ(V) and show that there exists two classes of maximal cliques in Γ(V). We also find the exact clique number of Γ(V) for some particular cases. Moreover, we provide some results on size, edge-connectivity and chromatic number of Γ(V).

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On a graph of ideals

Cited by 16 publications

References 14 publications

Annihilating-Ideal Graph of $C(X)$

Annihilating-Ideal Graph of $C(X)$

Graph theoretic representation of rings of continuous functions

On nonzero component graph of vector spaces over finite fields

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