The M-principal graph of a commutative ring

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by TΓ(R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of TΓ(R) where R is a commutative Artin ring. The intersection graph of gamma sets in TΓ(R) is denoted by ITΓ(R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory32 (2012) 339–354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph ITΓ (ℤn) of gamma sets in TΓ(ℤn). In this paper, we study about ITΓ(R), where R is a commutative Artin ring. Actually we investigate the interplay between graph-theoretic properties of ITΓ(R) and ring-theoretic properties of R. At the first instance, we prove that diam (ITΓ(R)) ≤ 2 and gr (ITΓ(R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of ITΓ(R) are given. Further, we discuss about the vertex-transitive property of ITΓ(R). At last, we obtain all commutative Artin rings R for which ITΓ(R) is either planar or toroidal or genus two.

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The annihilator graph of commutative ring [Formula: see text] is the simple undirected graph AG([Formula: see text]) with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal. In this paper, we classify all finite commutative rings with identity whose annihilator graph and essential graph have crosscap two.

Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that R ≅ Π 𝕑 2 n {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}} , n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

On the coloring of the annihilating-ideal graph of a commutative ring

<span>The notion of the matrix Jacobson graph was introduced in 2019. Let </span><span class="math inline"><em>R</em></span><span> be a commutative ring and </span><span class="math inline"><em>J</em>(<em>R</em>)</span><span> be the Jacobson radical of ring </span><span class="math inline"><em>R</em></span><span>. The matrix Jacobson graph of ring </span><span class="math inline"><em>R</em></span><span> size </span><span class="math inline"><em>m</em> × <em>n</em></span><span>, denoted </span><span class="math inline">𝔍(<em>R</em>)^{<em>m</em> × <em>n</em>}</span><span>, is defined as a graph where the vertex set is </span><span class="math inline"><em>R</em>^{<em>m</em> × <em>n</em>} ∖ <em>J</em>(<em>R</em>)^{<em>m</em> × <em>n</em>}</span><span> such that two distinct vertices </span><span class="math inline"><em>A</em>, <em>B</em></span><span> are adjacent if and only if </span><span class="math inline">1 − det(<em>A</em>^<em>t</em><em>B</em>)</span><span> is not a unit in ring </span><span class="math inline"><em>R</em></span><span>. Here we obtain some graph theoretical properties of </span><span class="math inline">𝔍(<em>R</em>)^{<em>m</em> × <em>n</em>}</span><span> including its connectivity, planarity and perfectness.</span>

Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

Let R be a commutative ring with identity and $$\mathbb {A}(R)$$ be the set of all annihilator ideals of R. The annihilator-ideal graph of R, denoted by $$A_I(R)$$ , is a simple graph with the vertex set $$\mathbb {A}^*(R) := \mathbb {A}(R)\setminus \{(0)\}$$ , and two distinct vertices I and J are adjacent if and only if $$ann(IJ)\ne ann(I) \cup ann(J)$$ . In this paper, we characterize all isomorphism classes of Artinian commutative rings whose $$A_I(R)$$ has genus one and crosscap one.

On the genus of nil-graph of ideals of commutative rings

Let R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

Let [Formula: see text] be a ring with unity. The cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertices are the set of all nonzero and non-unit elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. The reduced cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertex set is the set of all nontrivial principal ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Jesili et al. [On the genus of reduced cozero-divisor graph of commutative rings, Soft Comput. 27(2) (2023) 657–666] classified all non-local commutative rings whose reduced cozero-divisor graph is of genus one. In this paper, we characterize all classes of finite non-local commutative rings whose reduced cozero-divisor graph is of genus two. Moreover, we classify all the non-local commutative rings whose cozero-divisor graph is of genus two.

For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc (G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(R)\{0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are given.

Genus of total graphs from rings: A survey

Let$R$be a commutative ring. The regular digraph of ideals of$R$, denoted by$\Gamma (R)$, is a digraph whose vertex set is the set of all nontrivial ideals of$R$and, for every two distinct vertices$I$and$J$, there is an arc from$I$to$J$whenever$I$contains a nonzero divisor on$J$. In this paper, we study the connectedness of$\Gamma (R)$. We also completely characterise the diameter of this graph and determine the number of edges in$\Gamma (R)$, whenever$R$is a finite direct product of fields. Among other things, we prove that$R$has a finite number of ideals if and only if$\mathrm {N}_{\Gamma (R)}(I)$is finite, for all vertices$I$in$\Gamma (R)$, where$\mathrm {N}_{\Gamma (R)}(I)$is the set of all adjacent vertices to$I$in$\Gamma (R)$.

Comaximal graph of commutative rings