Geodesic
Unitary Invertible Graphs of Finite Rings
Discussiones Mathematicae. General Algebra and Applications, Tome 41 (2021) no. 1, pp. 195-208

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Let R be a finite commutative ring with unity. In this paper, we consider set of additive and mutual additive inverses of group units of R and obtain interrelations between them. In general φ(Zn) is even, however we demonstrate that φ(R) is odd for any finite commutative ring with unity of Char(R) ≠ 2. Further, we present unitary invertible graph related with self and mutual additive inverses of group units. At long last, we establish a formula for counting the total number of basic and non-basic triangles in the unitary invertible graph.
Keywords: finite commutative rings, additive and mutual additive inverses, Euler-function, unitary invertible graphs, basic and non-basic triangles 
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Chalapathi, Tekuri; Sajana, Shaik. Unitary Invertible Graphs of Finite Rings. Discussiones Mathematicae. General Algebra and Applications, Tome 41 (2021) no. 1, pp. 195-208. http://geodesic.mathdoc.fr/item/DMGAA_2021_41_1_a15/