The metric dimension of the total graph of a semiring

David Dolžan

doi:10.3842/umzh.v76i11.7980

David Dolžan Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Slovenia, and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

DOI: https://doi.org/10.3842/umzh.v76i11.7980

Ключові слова: english

Анотація

УДК 512.5

Метрична розмірність повного графа напівкільця

Обчислено метричну розмірність повного графа прямого добутку скінченних комутативних антинегативних напівкілець з їхніми множинами нульових дільників, замкнених при додаванні.

Посилання

F. Ali, M. Salman, S. Huang, On the commuting graph of dihedral group, Comm. Algebra, 44, № 6, 2389–2401 (2016). DOI: https://doi.org/10.1080/00927872.2015.1053488

D. F. Anderson, A. Badawi, The total graph of a commutative ring, J. Algebra, 320, № 7, 2706–2719 (2008). DOI: https://doi.org/10.1016/j.jalgebra.2008.06.028

F. Baccelli, J. Mairesse, Ergodic theorems for stochastic operators and discrete event networks, Idempotency (Bristol, 1994), Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 171–208 (1998). DOI: https://doi.org/10.1017/CBO9780511662508.011

P. J. Cameron, J. H. van Lint, Designs, graphs, codes and their links, London Math. Soc. Stud. Texts, vol. 22, Cambridge University Press, Cambridge (1991). DOI: https://doi.org/10.1017/CBO9780511623714

R. Cuninghame-Green, Minimax algebra, Lect. Notes Econ. and Math. Systems, vol. 166, Springer-Verlag, Berlin, New York (1979). DOI: https://doi.org/10.1007/978-3-642-48708-8

D. Dolžan, The metric dimension of the total graph of a finite commutative ring, Canad. Math. Bull., 59, № 4, 748–759 (2016). DOI: https://doi.org/10.4153/CMB-2016-015-5

D. Dolžan, The metric dimension of the annihilating-ideal graph of a finite commutative ring, Bull. Austral. Math. Soc., 103, № 3, 362–368 (2021). DOI: https://doi.org/10.1017/S0004972720001239

D. Dolžan, The metric dimension of the zero-divisor graph of a matrix semiring, Bull. Malays. Math. Sci. Soc., 46, № 6, 201 (2023). DOI: https://doi.org/10.1007/s40840-023-01591-2

D. Dolžan, P. Oblak, The total graphs of finite rings, Comm. Algebra, 43, № 7, 2903–2911 (2015). DOI: https://doi.org/10.1080/00927872.2014.907417

D. Dolžan, P. Oblak, The total graphs of finite commutative semirings, Results Math., 72, № 1-2, 193–204 (2017). DOI: https://doi.org/10.1007/s00025-016-0595-y

Sh. E. Atani, F. E. Kh. Saraei, The total graph of a commutative semiring, An. Sƫiinƫ. Univ. ``Ovidius'' Constanƫa Ser. Mat., 21, № 2, 21–33 (2013). DOI: https://doi.org/10.2478/auom-2013-0021

R. C. Entringer, L. D. Gassman, Line-critical point determining and point distinguishing graphs, Discrete Math., 10, 43–55 (1974). DOI: https://doi.org/10.1016/0012-365X(74)90019-3

F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Combin., 2, 191–195 (1976).

C. Hernando, M. Mora, I. M. Pelayo, C. Seara, D. R. Wood, Extremal graph theory for metric dimension and diameter, Electron. J. Combin., 17, № 1, Article 30 (2010). DOI: https://doi.org/10.37236/302

L. John, H. Vijay, On the total graph of the semiring of matrices over the Boolean semiring, Bull. Kerala Math. Assoc., 13, № 2, 199–207 (2016).

S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math., 70, № 3, 217–229 (1996). DOI: https://doi.org/10.1016/0166-218X(95)00106-2

P. Li, A heuristic method to compute the approximate postinverses of a fuzzy matrix, IEEE Trans. Fuzzy Systems, 22, 1347–1351 (2014). DOI: https://doi.org/10.1109/TFUZZ.2013.2282231

Shikun Ou, Dein Wong, Fenglei Tian, Q. Zhou, Fixing number and metric dimension of a zero-divisor graph associated with a ring, Linear and Multilinear Algebra, 69, № 10, 1789–1802 (2021). DOI: https://doi.org/10.1080/03081087.2020.1815639

S. Pirzada, M. Aijaz, S. P. Redmond, Upper dimension and bases of zero-divisor graphs of commutative rings, AKCE Int. J. Graphs Comb., 17, № 1, 168–173 (2020). DOI: https://doi.org/10.1016/j.akcej.2018.12.001

S. Pirzada, R. Raja, On the metric dimension of a zero-divisor graph, Comm. Algebra, 45, № 4, 1399–1408 (2017). DOI: https://doi.org/10.1080/00927872.2016.1175602

R. Raja, S. Pirzada, Sh. Redmond, On locating numbers and codes of zero divisor graphs associated with commutative rings, J. Algebra and Appl., 15, № 1, Article 1650014 (2016). DOI: https://doi.org/10.1142/S0219498816500146

Sh. Redmond, S. Szabo, When metric and upper dimensions differ in zero divisor graphs of commutative rings, Discrete Math. Lett., 5, 34–40 (2021). DOI: https://doi.org/10.47443/dml.2021.0005

A. Sebö, E. Tannier, On metric generators of graphs, Math. Oper. Res., 29, № 2, 383–393 (2004). DOI: https://doi.org/10.1287/moor.1030.0070

P. J. Slater, Leaves of trees, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), Congressus Numerantium, № 14, Utilitas Math., Winnipeg, Man. (1975), p. 549–559.

P. J. Slater, Fault-tolerant locating-dominating sets, vol. 249, Combinatorics, Graph Theory and Computing (Louisville, KY, 1999) (2002), p. 179–189. DOI: https://doi.org/10.1016/S0012-365X(01)00244-8

D. P. Sumner, Point determination in graphs, Discrete Math., 5, 179–187 (1973). DOI: https://doi.org/10.1016/0012-365X(73)90109-X

Shan Zhao, Xueping Wang, Invertible matrices and semilinear spaces over commutative semirings, Inform. Sci., 180, № 24, 5115–5124 (2010). DOI: https://doi.org/10.1016/j.ins.2010.08.033