In this article the annihilating-ideal graph of the ring C(X) is studied. We have tried to associate the graph properties of AG(X), the ring properties of C(X) and the topological properties of X. It is shown that X has an isolated point if and only if R is a direct summand of C(X) and this happens if and only if AG(X) is not triangulated. Radius, girth, dominating number and clique number of the AG(X) are investigated. It is proved that c(X) <= dt(AG(X)) ,= w(X) and wAG(X) = χAG(X) = c(X).
G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr, and F. Shaveisi, On the coloring of the annihilating-ideal graph of a commutative ring, Discrete Math., 312 (2012), 2620–2626.
F. Aliniaeifard and M. Behboodi, Rings whose annihilating-ideal graphs have positive genus, J. Algebra Appl., 11 (2012), 1250049.
A. Amini, B. Amini, E. Momtahan, and M. H. Shirdareh Haghighi, On a graph of ideals, Acta Math. Hung., 134 (2012), 369–384.
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Reading, Mass.-Menlo Park, Calif.-London-Don Mills, Ont.: Addison-Wesley Publishing Company, 1969.
F. Azarpanah and M. Motamedi, Zero-divisor graph of C(X), Acta Math. Hung., 108 (2005), 25–36.
M. Badie, Comaximal graph of C(X), Commentat. Math. Univ. Carol., 57 (2016), 353–364.
M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings. I., J. Algebra Appl., 10 (2011), 727–739.
M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings. II., J. Algebra Appl., 10 (2011), 741–753.
J.A. Bondy and U.S.R. Murty, Graph theory with application, The Macmillan Press, New York, 1976.
T. Chelvam and K. Selvakumar, On the connectivity of the annihilating-ideal graphs, Discuss. Math. Gen. Algebra Appl., 35 (2015), 195–204.
L. Gillman and M. Jerison, Rings of continuous functions, The University Series in Higher Mathematics. Princeton-Toronto-London-New York: D. Van Nostrand Company, Inc., 1960.
J. Guo, T. Wu, and H. Yu, On rings whose annihilating-ideal graphs are blowups of a class of Boolean graphs, J. Korean Math. Soc., 54 (2017), 847–865.
R. Nikandish and H. R. Maimani, Dominating sets of the annihilating-ideal graphs, Electron. Notes Discrete Math., 45 (2014) 17–22.
R. Nikandish, H. R. Maimani, and S. Kiani, Domination number in the annihilating-ideal graphs of commutative rings, Publ. Inst. Math., Nouv. Ser., 97 (2015), 225–231.
S. Willard, General topology., Addison-Wesley Series in Mathematics. Reading, Mass. Addison-Wesley Publishing Company., 1970.
)
