Abstract
In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail.
Highlights
We considered a generalization of dot total graph of R as well as an ideal-based zero-divisor graph
We showed that TI (Γ(R)) is connected and has a small diameter of at most two
The application of this graph to the study on Laplacian eigenvalues of an ideal-based dot total graph, which is closely related to the work in the paper [6], can be investigated
Summary
X and y are vertices that are both distinct and adjacent if and only if xy ∈ I, i.e., Γ I (R) is subgraph of TI (Γ(R)). The graph has vertices x and y that are both distinct and adjacent if and only if xy ∈ I. We will connect the edges between the vertices defined in the three previous sets as follows: We define a complete graph (Kn , where n = | I |) by using the first set I = ( a) as its vertex set.
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