**1. Introduction**

Let *Z*(R) and *Reg*(R) be a set of zero-divisors and a set of regular elements of commutative ring R with 1 = 0, respectively. In [1], Mohammad Ashraf et al. defined the dot total graph of R, denoted by *TZ*(R)(Γ(R)), as an (undirected) graph, which consists of all elements of R as vertex set *V*(*TZ*(R)(Γ(R))) and includes all edges such that for distinct *x*, *y* ∈ R, *e* = *xy* ∈ *E*(*TZ*(R)(Γ(R))) if and only if *xy* ∈ *Z*(R). In this paper, we replace *Z*(R) by an ideal *I*, and we introduce and investigate an *ideal-based dot total graph of* R denoted *TI*(Γ(R)). In addition, Redmond [2] defined Γ*I*(R) as an undirected graph. It has vertices {*x* ∈R\ *I* | *xy* ∈ *I f or some y* ∈R\ *I*}. In this case, *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)). It will also appear in this paper. Further, if *I* = (0) in Γ*I*(R), then Γ*I*(R) = Γ(R); this graph is studied by Anderson et al. [3], and they were interested in studying the interplay of ring-theoretic properties of R with graph-theoretic properties of Γ(R). Moreover, they associated a (simple) graph Γ(R) to R, which consists of a vertex set *V*(Γ(R)) = *Z*(R)<sup>∗</sup> = *Z*(R) \ {0} and edge set *E*(Γ(R)) such that for all distinct *x*, *y* ∈ *Z*(R)∗, *e* = *xy* ∈ *E*(Γ(R)) if and only if *xy* = 0. Furthermore, if *I* = (0) in *TI*(Γ(R)), then *TI*(Γ(R)) = Γ0(R); this graph is studied by Beck [4], in which he considered R as a simple graph for which its vertex set is the set of all elements of R and edge set such that for all distinct *x*, *y* ∈ R, *e* = *xy* ∈ *E*(Γ0(R)) if and only if *xy* = 0. In addition, some fundamentals of Laplacian eigenvalues and energy of graphs can be identified in [5–7].

Assuming *G* to be a graph, *G* can be said to be connected when a path connects every pair of its distinctive vertices. Denoting distinct vertices of graph *G* to be *x* and *y*, *d*(*x*, *y*) will indicate the shortest distance between the two vertices. However, where no such path exists, it will be represented by *d*(*x*, *y*) = ∞. Similarly, the diameter of *G* is *diam*(*G*) = *sup*{*d*(*x*, *y*) | *x and y are distinct vertices o f G*}. The girth of *G*, denoted by *gr*(*G*), is defined as the length of shortest cycle in *G* (*gr*(*G*) = ∞ if *G* contains no cycle). Note that if *G* contains a cycle, then *gr*(*G*) ≤ 2 *diam*(*G*) + 1. The degree of vertex *v*, written *degG*(*v*) or *deg*(*v*), is the number of edges incident to *v* (or the degree of the vertex *v* is the number of vertices adjacent to *v*). In a connected graph *G*, a vertex *v* is said to be a cut-vertex of *G* if and only if *G* \ {*v*} is disconnected. Let *V*(*G*) be a vertex set of *G*. Then, the subset *U* ⊆ *V*(*G*) is called a vertex-cut if *G* \ *U* is disconnected. The connectivity of a

**Citation:** Ashraf, M.; Asalool, J.H.; Alanazi, A.M.; Alamer, A. An Ideal-Based Dot Total Graph of a Commutative Ring. *Mathematics* **2021**, *9*, 3072. https://doi.org/10.3390/ math9233072

Academic Editors: Irina Cristea and Hashem Bordbar

Received: 6 November 2021 Accepted: 25 November 2021 Published: 29 November 2021

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graph *G* denoted by *k*(*G*) and is defined as the cardinality of a minimum vertex-cut of *G*, which is also the same concepts we have in the edges. In a connected graph *G*, an edge *e* is said to be a bridge of *G* if and only if *G* \ {*e*} is disconnected. Let *E*(*G*) be an edge set of *G*. If *G* \ *X* is disconnected, it will have a subset *X* ⊆ *E*(*G*) as its edge-cut. Let *λ*(*G*) denote the edge-connectivity of a connected graph *G* which is the size of the smallest set of edges for which removal disconnects *G*. Moreover, a clique is a complete subgraph of a graph *G*. The clique number denoted by *ω*(*G*) is the greatest integer *n* - 1 such that *Kn* ⊆ *G*, and *ω*(*G*) = ∞ if *Kn* ⊆ *G* for all *n* - 1. A nontrivial connected graph *G* is Eulerian if every vertex of *G* has an even degree. Moreover, *G* contains a Eulerian trail if exactly two vertices of *G* have an odd degree. In addition, let *G* be a graph of order *n* ≥ 3. If *deg*(*u*) + *deg*(*v*) ≥ *n* for each pair *u* and *v* of vertices of *G* that are not adjacent, then *G* is Hamiltonian.

The present paper is organized as follows:

In Section 2, we define an ideal-based dot total graph of R and study the most basic results of *TI*(Γ(R)). We provide many examples and show that *TI*(Γ(R)) is always connected with *diam*(*TI*(Γ(R))) 2 and *gr*(*TI*(Γ(R))) 5, and we determine when *TI*(Γ(R)) is a complete graph and a regular graph. Moreover, we find the degree of each vertex of *TI*(Γ(R)). Furthermore, in Section 3, we study the connectivity of *TI*(Γ(R)) when *TI*(Γ(R)) has a no cut-vertex, and *TI*(Γ(R)) has a bridge. We shall also find the *k*(*TI*(Γ(R))). On the other hand, in Section 4, we study the clique number and girth of *TI*(Γ(R)), and we determine the clique number when *TI*(Γ(R)) has a cycle. Furthermore, we find the girth of *TI*(Γ(R)), i.e., *gr*(*TI*(Γ(R))). Finally, in Section 5, we study the traversability of *TI*(Γ(R)) when *TI*(Γ(R)) is Eulerian or contains a Eulerian trail, and *TI*(Γ(R)) is Hamiltonian.
