Author:
Ansari-Toroghy H.,Habibi S.
Abstract
AbstractLet M be a module over a commutative ring R. In this paper, we continue our study about the Zariski topology-graph $$G(\tau _T)$$
G
(
τ
T
)
which was introduced in Ansari-Toroghy et al. (Commun Algebra 42:3283–3296, 2014). For a non-empty subset T of $$\mathrm{Spec}(M)$$
Spec
(
M
)
, we obtain useful characterizations for those modules M for which $$G(\tau _T)$$
G
(
τ
T
)
is a bipartite graph. Also, we prove that if $$G(\tau _T)$$
G
(
τ
T
)
is a tree, then $$G(\tau _T)$$
G
(
τ
T
)
is a star graph. Moreover, we study coloring of Zariski topology-graphs and investigate the interplay between $$\chi (G(\tau _T))$$
χ
(
G
(
τ
T
)
)
and $$\omega (G(\tau _T))$$
ω
(
G
(
τ
T
)
)
.
Funder
Institute for Research in Fundamental Sciences
Publisher
Springer Science and Business Media LLC
Reference17 articles.
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3. Ansari-Toroghy, H.; Farshadifar, F.: Product and dual product of submodules. Far East J. Math. Sci 25(3), 447–455 (2008)
4. Ansari-Toroghy, H.; Habibi, S.: The Zariski topology-graph of modules over commutative rings. Commun. Algebra 42, 3283–3296 (2014)
5. Ansari-Toroghy, H.; Habibi, S.: The annihilating-submodule graph of modules over commutative rings. Math. Rep. 20(70), 245–262 (2018)
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