Abstract
Let R be a commutative ring with non-zero identity and J(R) be Jacobson ideal of R. The Jacobson graph of R is the graph whose vertices are \(R{\setminus } J(R)\), and two different vertices x and y are adjacent if \(1-xy\notin U(R)\), where U(R) is the set of units of R. We investigate diameter of \(\mathfrak {J}_R\) and seek relation between it and diameter of Jacobson graphs under extension to polynomial and power series rings. Also, vertex and edge connectivity of finite Jacobson graphs are obtained. Finally, we show that all finite Jacobson graphs have a matching that misses at most one vertex and offer one 1-factor decomposition of a regular induced subgraph.
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Azimi, A., Erfanian, A., Farrokhi, D.G.M.: Isomorphisms between Jacobson graphs. Rend. Circ. Mat. Palermo 63, 277–286 (2014)
Azimi, A., Erfanian, A., Farrokhi, D.G.M.: The Jacobson graph of commutative rings. J. Algebra Appl. 12(3), 18 (2013)
Azimi, A., Farrokhi, D.G.M.: Cycles and paths in Jacobson graphs. To appear in Ars. Combinatoria (2016)
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)
Chartrand, G., Zhang, P.: Chromatic Graph Theory. Taylor & Francis, UK (2009)
Macdonald, B.R.: Finite Rings with Identity. Marcel Dekker Inc, New York (1974)
Watkins, J.J.: Topics in Commutative Ring Theory. Princeton University Press, Princeton (2007)
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The authors would like to thank the referee for some helpful comments and suggestions that improved the paper substantially.
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Ghayour, H., Erfanian, A. & Azimi, A. Some results on the Jacobson graph of a commutative ring. Rend. Circ. Mat. Palermo, II. Ser 67, 33–41 (2018). https://doi.org/10.1007/s12215-016-0290-6
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DOI: https://doi.org/10.1007/s12215-016-0290-6