The paper studies the structures of the total and regular graphs of sets of three elements over fields of zero characteristic. The graphs themselves are classified up to isomorphism.
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S. Akbari, M. Jamaali, and S. A. Seyed Fakhari, “The clique numbers of regular graphs of matrix algebras are finite,” Linear Algebra Appl., 431, 1715–1718 (2009).
D. F. Anderson and A. Badawi, “The total graph of a commutative ring,” J. Algebra, 320, 2706–2719 (2008).
S. Akbari, M. Aryapoor, and M. Jamaali, “Chromatic number and clique number of subgraphs of regular graph of matrix algebras,” Linear Algebra Appl., 436, 2419–2424 (2012).
S. Akbari and F. Heydari, “The regular graph of a noncommutative ring,” Bull. Austral. Math. Soc., 89, No. 1, 132–140 (2014).
E. I. Bunina, A. V. Mikhalev, and A. G. Pinus, Elementary and Other Relative Logical Equivalences of Classical Universal Algebras [in Russian], MCNMO, Moscow (2015).
P. J. Cameron, “Research problems from the BCC22,” Discrete Math., 311, 1074–1083 (2011).
A. E. Guterman, A. M. Maksaev, and V. V. Promyslov, “Pairs of maps preserving singularity on subsets of matrix algebras,” Linear Algebra Appl., 644, 1–27 (2022).
F. Harary, Graph Theory [Russian translation], Mir, Moscow (1973).
U. Knauer and K. Knauer, Algebraic Graph Theory: Morphisms, Monoids and Matrices, 2nd rev. and ext. ed., de Gruyter (2015).
A. M. Maksaev and V. V. Promyslov, “On total and regular graphs of a polynomial,” Fundam. Prikl. Mat., 23, No. 4, 113–142 (2021).
V. V. Promyslov, “Classification of regular graphs of three-point sets,” Intel. Systems. Theory Appl., 25, No. 4, 205–208 (2021).
I. Tomon, “On the chromatic number of regular graphs of matrix algebras,” Linear Algebra Appl., 475, 154–162 (2015).
J. Zhou, D. Wong, and X. Ma, “Automorphism group of the total graph over a matrix ring,” Linear Multilinear Algebra, 65, No. 3, 572–581 (2017).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 514, 2022, pp. 167–192.
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Promyslov, V.V. Classification of the Total and Regular Graphs of Three-Point Sets. J Math Sci 272, 592–607 (2023). https://doi.org/10.1007/s10958-023-06452-8
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DOI: https://doi.org/10.1007/s10958-023-06452-8