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In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

For a finite noncyclic group G, let Cyc (G) be a set of elements a of G such that ⟨ a , b ⟩ is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set G \ Cyc (G) , having an edge between two distinct vertices x and y if ⟨ x , y ⟩ is not cyclic. In this paper, we show that, for a fixed nonnegative integer k, there are at most finitely many finite noncyclic groups whose noncyclic graphs have (non)orientable genus k. We also classify the finite noncyclic groups whose noncyclic graphs have (non)orientable genus 1, 2, 3 and 4, respectively. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by ΓRn, with Rn\{0} as the vertex set and two distinct vertices X and Y in Rn being adjacent if and only if there exists an n×n lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that XTAY=0 or YTAX=0, where, for a matrix B,BT is the matrix transpose of B. If n=1, then ΓRn is isomorphic to the zero divisor graph Γ(R), and so ΓRn is a generalization of Γ(R) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of ΓRn. We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three. [ABSTRACT FROM AUTHOR]