335 KB41 KB200820252022Nazim, MohdRehman, Nadeem Urletnik:22številka:3P3.05 (16 str.)DOI:10.26493/1855-3974.2645.8fcCOBISSID_HOST:118098435ISSN:1855-3966URN:URN:NBN:SI:doc-JMN5XPYMenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaannihilating-ideal graphcomplete graphgenus of a graphgraf anihilirajočih idealovgraf deliteljev ničaplanar graphpolni grafravninski grafrod grafazero-divisor graphOn the essential annihilating-ideal graph of commutative rings|Let ?$R$? be a commutative ring with unity, ?$A(R)$? be the set of all annihilating-ideals of ?$R$? and ?$A^{\ast}(R) = A(R) \setminus \{0\}$?. In this paper, we introduced and studied the essential annihilating-ideal graph of ?$R$?, denoted by ?$\mathcal{EG}(R)$?, with vertex set ?$A^{\ast}(R)$? and two distinct vertices ?$I_1$? and ?$I_2$? are adjacent if and only if v ?$Ann (I_1 I_2)$? is an essential ideal of ?$R$?. We prove that ?$\mathcal{EG}(R)$? is a connected graph with diameter at most three and girth at most four if ?$\mathcal{EG}(R)$? contains a cycle. Furthermore, the rings ?$R$? are characterized for which ?$\mathcal{EG}(R)$? is a star or a complete graph. Finally, we classify all the Artinian rings ?$R$? for which ?$\mathcal{EG}(R)$? is isomorphic to some well-known graphsNaj bo ?$R$? komutativen kolobar z enoto, ?$A(R)$? množica anihilirajočih idealov kolobarja ?$R$? in ?$A^{\ast}(R) = A(R) \setminus \{0\}$?. V tem članku vpeljemo in preučujemo esencialni graf anihilirajočih idealov kolobarja ?$R$?, označen z ?$\mathcal{EG}(R)$?; množica njegovih vozlišč je ?$A^{\ast}(R)$?, dva različna vozlišča ?$I_1$? in ?$I_2$? tega grafa pa sta sosedna če in samo če je ?$Ann (I_1 I_2)$? esencialni ideal kolobarja ?$R$?. Dokažemo, da je ?$\mathcal{EG}(R)$? povezan graf s premerom največ tri in ožino največ štiri, če ?$\mathcal{EG}(R)$? vsebuje cikel. Karakteriziramo kolobarje ?$R$?, za katere je ?$\mathcal{EG}(R)$? zvezda ali pa polni graf. Klasificiramo tudi vse Artinske kolobarje ?$R$?, za katere je ?$\mathcal{EG}(R)$? izomorfen nekaterim znanim grafomTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije