Total Graph Of Commutative Ring Research Articles

An Ideal-Based Dot Total Graph of a Commutative Ring

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.

Bounds for the Genus of Generalized Total Graph of a Commutative Ring

Let R be a commutative ring with non-zero identity and I its proper ideal. The total graph of R with respect to I, denoted by T (ΓI (R)), is the undirected graph with all elements of R as vertices, and where distinct vertices x and y are adjacent if and only if [Formula: see text]. In this paper, some bounds for the genus of T(ΓI(R)) are obtained. We improve and generalize some results for the total graphs of commutative rings. In addition, we obtain an isomorphism relation between two ideal based total graphs.

Genus of total graphs of commutative rings: A survey

The topological properties of graphs are well studied. More specifically the genus of graphs and their embeddings are attractions to many authors. In this survey, we present the results concerning the genus of total graph and generalized the total graph and generalized total graphs of commutative rings.

𝒞-Consistency in signed total graphs of commutative rings

Motivated by the earlier study on the notion of signed total graph of a commutative ring, in this paper, we characterize all the commutative rings with unity for which signed total graph is [Formula: see text]-consistent and sign-compatible. To do this, first, we derive a formula to determine the degree of each vertex in [Formula: see text] (induced subgraph of the total graph), when [Formula: see text] and [Formula: see text], where each [Formula: see text]’s is a field of characteristic [Formula: see text], [Formula: see text] is an odd prime, [Formula: see text] and [Formula: see text].

Balanced Signed Total Graphs of Commutative Rings

This paper is primarily concerned with a natural extension of the notion of a total graph $$T(\varGamma (R))$$T(Γ(R)) in the realm of signed graph for a finite commutative ring R. First, we characterize the rings for which the signed total graph $$T_{\varSigma }(\varGamma (R))$$TΣ(Γ(R)) and its line signed graph $$L(T_{\varSigma }(\varGamma (R)))$$L(TΣ(Γ(R))) are balanced. Second, we characterize the rings for which the negation of a signed total graph $$\eta (T_{\varSigma }(\varGamma (R)))$$?(TΣ(Γ(R))) is balanced. Several new directions for further research are also indicated.

Calculating different topological indices of total graph of ℤn

A recently published paper [6] considered the total graph of commutative ring R. In this paper, we compute Wiener, hyper-Wiener, reverse Wiener, Randić, Zagreb, ABC and GA indices of zero-divisor graph.

Projective total graphs of commutative rings

Projective total graphs of commutative rings