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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]

The intersection power graph of a finite group GG is the graph whose vertex set is GG, and two distinct vertices xx and yy are adjacent if either one of xx and yy is the identity element of GG, or ⟨x⟩∩⟨y⟩\langle x\rangle \cap \langle y\rangle is non-trivial. In this paper, we completely classify all finite groups whose intersection power graphs are toroidal and projective-planar.

Let R be a finite commutative ring with identity. The idempotent graph of R is the simple undirected graph I(R) with vertex set, the set of all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we have determined all isomorphism classes of finite commutative rings with identity whose I(R) has genus one or two. Also we have determined all isomorphism classes of finite commutative rings with identity whose I(R) has crosscap one. Also we study the the book embedding of toroidal idempotent graphs and classify finite commutative rings whose I(R) is a ring graph.