Abstract
Let R be a commutative ring and let \({n >1}\) be an integer. We introduce a simple graph, denoted by \({\Gamma_t(M_n(R))}\), which we call the trace graph of the matrix ring \({M_n(R)}\), such that its vertex set is \({M_n(R)^{\ast}}\) and such that two distinct vertices A and B are joined by an edge if and only if \({{\rm Tr} (AB)=0}\) where \({ {\rm Tr} (AB)}\) denotes the trace of the matrix AB. We prove that \({\Gamma_t(M_n(R))}\) is connected with \({{\rm diam}(\Gamma_{t}(M_{n}(R)))=2}\) and \({{\rm gr} (\Gamma_t(M_n(R)))=3}\). We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of \({\Gamma_t(M_n(R))}\). Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal.
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Akgünes, N.: Analyzing special parameters over zero-divisor graphs. AIP Conference Proceedings 1479, 390–393 (2012)
Amini, A., Amini, B., Momtahan, E., Shirdareh, M.H.: Haghighi, On a graph of ideals. Acta Math. Hungar. 134, 369–384 (2012)
Anderson, D.F.: On the diameter and girth of a zero divisor graph. II. Houston J. Math. 34, 361–371 (2008)
Anderson, D.F., Badawi, A.: On the zero-divisor graph of a ring. Comm. Algebra 36, 3073–3092 (2008)
D. F. Anderson, M. Axtell and J. Stickles, Zero-divisor graphs in commutative rings, in: Commutative Algebra, Noetherian and Non-Noetherian Perspectives, M. Fontana, S.E. Kabbaj, B. Olberding, I. Swanson, editors, Springer-Verlag (New York, 2010), pp. 23–45
Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)
Ashrafi, N., Maimani, H.R., Pournaki, M.R., Yassemi, S.: Unit graphs associated with rings. Comm. Algebra 38, 2851–2871 (2010)
Beck, I.: Coloring of commutative rings. J. Algebra 116, 208–226 (1988)
Bozic, I.I., Petrovic, Z.: Zero-divisor graphs of matrices over commutative rings. Comm. Algebra 37, 1186–1192 (2009)
S. Kabbaj and A. Mimouni, Zero-divisor graphs of amalgamations, Math. Scand. (To appear)
I. Kaplansky, Commutative Rings (rev. ed.), Univ. of Chicago Press (1974)
T. Y. Lam, A First Course in Noncommutative Rings, Springer Science Business Media (Berlin–Heidelberg–New York, 1999)
Li, A., Tucci, R.P.: Zero divisor graphs of upper triangular matrix rings. Comm. Algebra 41, 4622–4636 (2013)
Miguel, C.: Balanced zero-divisor graphs of matrix rings. Lobachevskii J. Math. 34, 137–141 (2013)
Mukwembi, S.: A note on diameter and the degree sequence of a graph. Appl. Math. Lett. 25, 175–178 (2012)
Mulay, S.B.: Cycles and symmetries of zero-divisors. Comm. Algebra 30, 3533–3558 (2002)
Okon, J.S.: Numbers of generators of ideals in a group ring of an elementary Abelian p-group. J. Algebra 224, 1–22 (2000)
Redmond, S.: The zero-divisor graph of a noncommutative ring. International J. Commutative Rings 1, 203–211 (2002)
Rush, D.: Rings with two-generated ideals. J. Pure Appl. Algebra 73, 257–275 (1991)
Rush, D.: Two-generated ideals and representations of abelian groups over valuation rings. J. Algebra 177, 77–101 (1995)
Smith, Z.O.: Planar zero-divisor graphs. Int. J. Commut. Rings 2, 177–188 (2003)
D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall (2001)
Wu, T., Yu, H., Lu, D.: The structure of finite local principal ideal rings. Comm. Algebra 40, 4727–4738 (2012)
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Almahdi, F.A.A., Louartiti, K. & Tamekkante, M. The trace graph of the matrix ring over a finite commutative ring. Acta Math. Hungar. 156, 132–144 (2018). https://doi.org/10.1007/s10474-018-0815-x
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DOI: https://doi.org/10.1007/s10474-018-0815-x