caa a22 4500 606200797 CHVBK 20210128100945.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s00025-015-0434-6 doi (NATIONALLICENCE)springer-10.1007/s00025-015-0434-6 On Some Graphs Associated to Commutative Semirings [Elektronische Daten] [Elham Mehdi-Nezhad, Amir Rahimi] Similar to the case of commutative rings, we define the notion of a zero-divisor graph of a commutative semiring R with respect to an ideal I of R, denoted by $${\Gamma_I(R)}$$ Γ I ( R ) . That is an undirected graph whose vertex set is the set $${\{x \in R {\setminus} I | xy \in I {\rm for some} y \in R {\setminus} I\}}$$ { x ∈ R \ I | x y ∈ I for some y ∈ R \ I } with distinct vertices x and y adjacent if and only if xy ∈ I. We discuss when $${\Gamma_I(R)}$$ Γ I ( R ) is r-partite. We also give some results on the subgraphs and the parameters of $${\Gamma_I(R)}$$ Γ I ( R ) . In addition, we apply these results to the annihilating-ideal graph of R with respect to I, denoted by $${\mathbb{AG}_I(R)}$$ AG I ( R ) , as an example (special case) of $${\Gamma_I(R)}$$ Γ I ( R ) . It is also shown that for I a radical ideal of $${R, \Gamma_I(R)}$$ R , Γ I ( R ) is isomorphic to a subgraph of $${\mathbb{AG}_I(R)}$$ AG I ( R ) which naturally asserts some known results (properties) between them interchangeably. Finally, we provide some counterexamples and construct a semiring whose ideal-based zero-divisor graph has a cut-point and more than one bridge, which is in contrast to the ring case. Springer Basel, 2015 Annihilating-ideal nationallicence semiring nationallicence zero-divisor nationallicence graph (with respect to an ideal) nationallicence r -partite graph nationallicence clique number nationallicence girth nationallicence cut-point nationallicence bridge nationallicence Mehdi-Nezhad Elham Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town, South Africa aut Rahimi Amir School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran aut Results in Mathematics Springer Basel 68/3-4(2015-11-01), 293-312 1422-6383 68:3-4<293 2015 68 25 https://doi.org/10.1007/s00025-015-0434-6 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00025-015-0434-6 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Mehdi-Nezhad Elham Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town, South Africa aut NATIONALLICENCE 700 1- Rahimi Amir School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran aut NATIONALLICENCE 773 0- Results in Mathematics Springer Basel 68/3-4(2015-11-01), 293-312 1422-6383 68:3-4<293 2015 68 25