Throughout this paper let R be a commutative ring identity, I be a proper ideal of R which is not a prime ideal of R, Z(R) be the set of zero-divisors of R,Z∗(R)=Z(R)∖{0},ZI∗(R)={u∉I | uv∈I for some v∉I}, ZI(R)=ZI∗(R)∪I and N(R) be the set of nilpotent elements of R. Let B be a submodule of an R-module M and X be any subset of M. Then (B:X)={r∈R | rx∈B for all x∈X}.MinI(R) will denote the set of minimal prime ideals of R which contain I. Let β(I)={r∈R | rn∈I for some n∈ℕ} be a prime radical of I in R, then β∗(I)=β(I)∖I and I is said to be radical ideal if β(I)=I.R/I denotes the quotient ring of R, and for any x+I∈R/I we use the notation X. For any subset A of R, we have A∗=A∖{0}.

Let G=(V(G),E(G)) be a graph, where V(G) denotes the set of vertices and E(G) be the set of edges of G. We say that G is connected if there exists a path between any two distinct vertices of G. For vertices a and b of G, d(a,b) denotes the length of the shortest path from a to b. In particular, d(a,a)=0 and d(a,b)=∞ if there exists no such path. The diameter of G, denoted by diam(G)=sup{d(a,b) | a,b∈V(G)}. A cycle in a graph G is a path that begins and ends at the same vertex. The girth of G, denoted by gr(G), is the length of a shortest cycle in G, (gr(G)=∞ if G contains no cycle). A complete graph G is a graph where all distinct vertices are adjacent. The complete graph with |V(G)|=n is denoted by Kn. A graph G is said to be complete k-partite if there exists a partition ∪ i=1k=Vi=V(G), such that u−v∈E(G) if and only if u and v are in different part of partition. If |Vi|=ni, then G is denoted by Kn1,n2,⋯,nk and in particular G is called complete bipartite if k=2. K1,n is said to be a star graph. G¯ denotes the complement graph of G. A graph H=(V(H),E(H)) is said to be a subgraph of G, if V(H)⊆V(G) and E(H)⊆E(G). Moreover, H is said to be induced subgraph of G if V(H)⊆V(G) and E(H)={u−v∈E(G) | u,v∈V(H)} and is denoted by G[V(H)]. Let H1 and H2 be two disjoint graphs. The join of H1 and H2, denoted by H1∨H2, is a graph with vertex set V(H1∨H2)=V(H1)∪V(H2) and edge set E(H1∨H2)=E(H1)∪E(H2)∪{u−v | u∈V(H1),v∈V(H2)}. Also G is called a null graph if E(G)=ϕ. For a graph G, a complete subgraph of G is called a clique. The clique number, ω(G), is the greatest integer n⩾1 such that Kn⊆G, and ω(G)=∞ if Kn⊆G for all n⩾1. The chromatic number χ(G) of a graph G is the minimum number of colours needed to colour all the vertices of G such that every two adjacent vertices get different colours. A graph G is perfect if χ(H)=ω(H) for every induced subgraph H of G. For a connected graph G,δ(G)=min{deg(x) | x∈V(G)},V(δ(G))={x | x∈V(G),deg(x)=δ(G)} and Δ(G)=max{deg(x) | x∈V(G)},V(Δ(G))={x | x∈V(G),deg(x)=Δ(G)}. A subset D⊆V(G) is said to be a dominating set if every vertex in V(G)∖D is adjacent to a vertex in D. A dominating set D is called a weak (or strong) dominating set if for every u∈V(G)∖D there exists v∈D with deg(v)≤deg(u) (or deg(u)≤deg(v)) and u is adjacent to v. The domination number γ(G) of G is defined to be minimum cardinality of a dominating set of G and such a dominating set of G is called a γ-set of G. In a similar way, we define the weak (or strong) domination number γw(G) (or γs(G)) of G. A graph G is said to be excellent, if for every u∈V(G), there exists a γ-set D containing u. Graph-theoretic terms are presented as they appear in Diestel [9].

The study of algebraic structure associated with graph is an active and interesting area of research. Several authors have done a lot of work in this area for instance, see [1, 2, 3, 7, 8, 12, 4]. The idea of a zero-divisor graph of a commutative ring {R} with identity was introduced by I. Beck in [8], who defined the graph on the vertex set {R} in which distinct vertices u,v ∈ {R} are adjacent if and only if uv=0. He was mainly interested in coloring of rings. The first simplification of Beck's zero divisor graph was introduced by Anderson and Livingston in [2]. We recall from [2] that a zero-divisor graph Γ(R) of R is the (undirected) graph with set of vertices Z∗(R) and on the vertex set Z∗(R), in which any two distinct vertices u and v of Γ(R) are adjacent if and only if uv=0. In [13] Redmond introduced an ideal-based zero-divisor graph ΓI(R) with set of vertices ZI∗(R) and vertex set ZI∗(R), in which any two distinct vertices u and v of ΓI(R) are adjacent if and only if uv∈I. In [5] Bakhtyiari et al. introduced the extended zero-divisor graph Γ′(R). The extended zero-divisor graph of R is an (undirected) graph Γ′(R) with the vertex set Z∗(R) and two distinct vertices u and v of Γ′(R) are adjacent if and only if either Ru∩annR(v)≠{0} or Rv∩annR(u)≠{0}.

In this paper we generalize the extended zero divisor graph Γ′(R) to an ideal-based extended zero-divisor graph ΓI′(R). The ideal based extended zero-divisor graph ΓI′(R) is the (undirected) graph with the vertex set ZI∗(R), in which two distinct vertices u and v are adjacent if and only if either (Ru+I)∩(I:{v})≠I or (Rv+I)∩(I:{u})≠I. If we take I=(0), then ΓI′(R)=Γ′(R). It follows that the ideal-based zero-divisor graph ΓI(R) is a subgraph of ΓI′(R). We prove that ΓI′(R) is connected with diameter at most two, and if ΓI′(R) contain a cycle, then girth is at most four. Furthermore, we study the connection between the ideal based extended zero-divisor graph ΓI′(R) and the ideal-based zero-divisor graph ΓI(R) associated with the ideal I of a commutative ring R. Among the other things, for a radical ideal of a commutative ring R, we show that ideal-based extended zero-divisor graph ΓI′(R) is identical to the ideal-based zero-divisor graph ΓI(R) if and only if R has exactly two minimal prime-ideals which contain I.

In this section, we generalize the notion of an extended zero-divisor graph Γ′(R) to an ideal-based extended zero-divisor graph ΓI′(R) and study fundamental properties of ΓI′(R).

Definition 2.1. Let I be an ideal in a commutative ring R with unity. An ideal-based extended zero divisor graph ΓI′(R) is an undirected graph with the set of vertices ZI∗(R), where any two distinct vertices u, v of ΓI′(R) are adjacent if and only if either (Ru+I)∩(I:{v})≠I or (Rv+I)∩(I:{u})≠I.

Proposition 2.2. Let {I} be an ideal in a commutative ring R with unity. Then

(i) ΓI(R) is a subgraph of ΓI′(R).

(ii) if I=(0), then ΓI′(R)=Γ′(R) and Γ(R) is a subgraph of ΓI′(R).

Proof. Let I be an ideal of a commutative ring R.

(i) Clearly, V(ΓI′(R))=V(ΓI(R)) and let u and v be any two adjacent vertices of ΓI(R). Then uv∈I and u∈(Ru+I)∩(I:{v}), v∈(Rv+I)∩(I:{u}), i.e., (Ru+I)∩(I:{v})≠I, (Rv+I)∩(I:{u})≠I. Hence u and v also adjacent in ΓI′(R), and by definition ΓI(R) is a subgraph of ΓI′(R).

(ii) It trivially holds.

Lemma 2.3. Let I be a radical ideal in a commutative ring R which is not prime, and let u∈ZI∗(R). Then

(i) (I:{u})=(I:{un}) for each positive integer n⩾2,

(ii) (Ru+I)∩(I:{u})=I.

Proof. Assume that I is a radical ideal of a ring R which is not prime and u∈ZI∗(R).

(i) Let n≥2. It is clear that (I:{u})⊆(I:{un}). If v∈(I:{un}), then vun∈I. Since I is a radical ideal, vu∈I and v∈(I:{u}). Thus (I:{un})=(I:{u}).

(ii) This is clearly true.

The following lemma gives several useful properties of ΓI′(R) and plays an important role in this section.

Lemma 2.4. Let I be a proper ideal of a ring R.

(i) If u-v is not an edge of ΓI′(R) for some u,v∈ZI∗(R), then (I:{u})=(I:{v}). If I is a radical ideal, then the converse is also true.

(ii) If (I:{u})⊈(I:{v}) or (I:{v})⊈(I:{u}) for some u,v∈ZI∗(R), then u-v is an edge of ΓI′(R).

(iii) If (Ru+I)∩(I:{u})≠I for some u∈ZI∗(R), then u is adjacent to all other vertex in ΓI′(R). In particular if u∈β∗(I), then u is adjacent to every other vertex of ΓI′(R).

(iv) ΓI′(R)[β∗(I)] is a complete subgraph of ΓI′(R).

Proof. Assume that I is an ideal of a ring R.

(i) If u-v is not an edge of ΓI′(R) for some u,v∈ZI∗(R), then (Ru+I)∩(I:{v})=I and (Rv+I)∩(I:{u})=I. Thus (Ru+I)(I:{v})⊆(Ru+I)∩(I:{v})=I and (Rv+I)(I:{u})⊆(Rv+I)∩(I:{u})=I and hence (I:{u})=(I:{v}). If I is a radical ideal of R, then by Lemma 2.3(ii), (Ru+I)∩(I:{v})=(Ru+I)∩(I:{u})=I and (Rv+I)∩(I:{u})=(Rv+I)∩(I:{v})=I. Thus u-v is not an edge of ΓI′(R).

(ii) This is clear by part (i).

t(iii) Assume that (Ru+I)∩(I:{u})≠I for some u∈ZI∗(R), and let v be another vertex of ΓI′(R). If u is not adjacent to v, then by part (i),(I:{u})=(I:{v}) and hence (Ru+I)∩(I:{u})=I, a contradiction.

(iv) This is clearly true by (iii).

Theorem 2.5. Let I be an ideal of R. Then ΓI′(R) is connected and dia(ΓI′(R))≤2. Moreover if ΓI′(R) contains a cycle, then gr(ΓI′(R))≤4.

Proof. By Lemma 2.2(i), ΓI(R) is a connected subgraph of ΓI′(R) such that V(ΓI(R))=V(ΓI′(R). Therefore ΓI′(R) is connected and gr(ΓI′(R))≤4. Now we prove that dia(ΓI′(R))≤2. If I is a non-radical ideal of R, then β(I)≠I and by Lemma 2.4(iii), dia(ΓI′(R))≤2. If I is a radical ideal of R, then β(I)=I. Let u,v∈V(ΓI′(R)) such that d(u,v)≠1. Then by Lemma 2.4(i), (I:{u})=(I:{v}). Since β(I)=I, by Lemma 2.3(ii), (Rv+I)∩(I:{v})=I. Therefore, for every w∈(I:{v})∖I both u,v are adjacent to w and d(u,v)=2. Thus diam(ΓI′(R))≤2. This completes the proof.

Lemma 2.6. Let I be a proper ideal of a commutative ring R. Then ZI(R) is a union of prime ideals of R which contain I.

Proof. Let us define a map F:R→R/I by F(x)=[x]. Clearly, F is a homomorphism from R onto R/I. By [11, p. 3], Z(R/I)=∪Pi where Pi is a prime ideal in R/I. clearly, ZI(R)=∪F−1(Pi) where F−1(Pi) is a prime ideal in R which contains I.

Corollary 2.7. Let I be a radical ideal of a commutative ring R. Then ZI(R)=∪Pi, where Pi∈MinI(R).

Proof. The corollary is immediate from Lemma 2.6 and [10, Corollory 2.4].

Theorem 2.8. Let I be a proper ideal of a commutative ring R and let ΓI′(R) contain a cycle. Then gr(ΓI′(R))=4 if and only if I is a radical ideal with |MinI(R)|=2.

Proof. First assume that gr(ΓI′(R))=4. If I is not a radical ideal, then β(I)≠I and by Lemma 2.4(iii) gr(ΓI′(R))=3, a contradiction. Hence I must be a radical ideal of R. Let u∈ZI∗(R). We will prove that (I:{u}) is a prime ideal of R. Suppose that ab∈ ({I}:{u}) such that a,b∉(I:{u}) but aubu∈I. Hence for every c∈(I:{u})∖I, it is easy to see that c-au-bu-c is a triangle, a contradiction. Hence (I:{u}) is a prime ideal. Since I is a radical ideal and by Lemma 2.3(ii) together with[10, Theorem 2.1] implies that (I:{u}) is a minimal prime ideal which contains I. i.e., (I:{u})∈MinI(R). By similar arguments (I:{v})∈MinI(R), for each v∈(I:{u})∖I. Now we prove that MinI(R)={(I:{u}),(I:{v})}. It is sufficient to show that (I:{u})∩(I:{v})=I. Assume on contrary (I:{u})∩(I:{v})≠I and a∈(I:{u})∩(I:{v})∖I. Then a-u-v-a is a triangle as uv∈I, a contradiction. Hence MinI(R)={(I:{u}),(I:{v})}.

Conversely, assume that I is a radical ideal of R and |MinI(R)|=2. Let Q1,Q2∈MinI(R). Since I is a radical ideal, we have ZI(R)=Q1∪Q2 and Q1∩Q2=I, by Corollary 2.7. It is not difficult to check that ΓI′(R) =K|Q1∗|,|Q2∗|, where |Q1∗|=|Q1∖I| and |Q2∗|=|Q2∖I|. Since ΓI′(R) contains a cycle, gr(ΓI′(R))=4.

Example 2.9. For R=ℤ6×ℤ3 and I=(0)×ℤ3, it may be observed that Q1=(3)×ℤ3 and Q2=(2)×ℤ3 are the only two minimal prime ideals of R, which contain radical ideal I, where ZI(R)=Q1∪Q2 and Q1∩Q2=I. Since |Q1∗|=3 and |Q2∗|=6, it can be easily seen in the following Figure 2.1 that ΓI′(R)=ΓI(R)= K|Q1∗|,|Q2∗|=K3,6 and gr(ΓI′(R))=4.

Figure 1. .

Example 2.10. For R=ℤ2×ℤ2×ℤ2 and I=(0)×(0)×ℤ2, it can be easily seen in the above Figure 2.2, K2,2 is realizable as ΓI′(R), which is not realizable as Γ′(R).

Figure 2. .

Corollary 2.11. Let I be a proper ideal of a commutative ring R. Then ΓI′(R) is K2,2 if and only if I is a radical ideal of R with |MinI(R)|=2 and each element of MinI(R) contains exactly two elements other than I.

Example 2.12. For R=ℤ24 and I=(8), it can be easily seen from the following Figures 2.3 and 2.4 that the ideal-based extended zero divisor graph ΓI′(R)=K9 is different from ideal-based zero divisor graph ΓI(R) and ΓI(R) is a subgraph of ΓI′(R)=K9.

Figure 3. .

As we have seen in the previous section, ideal-based extended zero divisor graphs and ideal-based zero-divisor graphs are close to each other, it would be interesting to characterize ideals of a ring whose ideal-based extended zero-divisor graph and ideal-based zero divisor graph are identical. We first study the case when I is a radical ideal of R.

Theorem 3.1. Let I be a radical ideal of a commutative ring R with |MinI(R)|=k≥2. Then k=2 if and only if ΓI′(R)=ΓI(R).

Proof. First assume that ΓI′(R)=ΓI(R). To prove that k=2, assume on the contrary Q1,Q2,Q3 are distinct minimal prime ideals of R which contain I. Let u∈Q1∖Q2∪Q3. Thus Q2∪Q3⊈(I:{u}) as (I:{u})⊆Q2∩Q3. So one may choose uv∉I, for some v∈Q2∪Q3∖Q1. Without loss of generality, assume that v∈Q2∖Q1. Obviously, (I:{v})⊆Q1. Also, it follows from [10, Theorem 2.1], there exists an element w∈(I:{u}) such that w∉Q1. Therefore, (I:{u})≠(I:{v}) and by Theorem 2.4(ii), u-v is an edge of ΓI′(R), a contradiction.

Conversely, assume that Q1 and Q2 are only two distinct minimal prime ideals of R which contain I. It is not difficult to check that ΓI(R)=ΓI′(R)=K|Q1∗|,|Q2∗|. Where Q1∗=Q1∖I and Q2∗=Q2∖I.

The following corollary follows from Theorem 3.1.

Corollary 3.2. Let I be a radical ideal of a commutative ring R, which is not a prime ideal. Then the following statements are equivalent:

(i) gr(ΓI′(R))=4.

(ii) ΓI′(R)=ΓI(R) and gr(ΓI(R))=4.

(iii) |MinI(R)|=2 and each minimal prime ideal of MinI(R) has at least two different elements other then elements of I.

(iv) ΓI′(R)=Km,n for some m,n∈ℕ and m,n⩾2.

In the rest of this section we study the case that I is a non radical ideal of R

Theorem 3.3. Let I be a non radical ideal of a commutative ring R. Then the following statements are equivalent.

(i) ΓI′(R)=ΓI(R).

(ii) If uv∉I for some u,v∈ZI∗(R), then (I:{u})=(I:{v}) and (I:{u}) is a prime ideal of R.

Proof. (i)⇒(ii) Assume that uv∉I, for some u,v∈ZI∗(R). Since ΓI′(R)=ΓI(R), we deduce that (I:{u})=(I:{v}), by Lemma 2.4(i). We now show that (I:{u}) is a prime ideal of R. Let ab∈(I:{u}), a∉(I:{u})and b∉(I:{u}). Then au∉I and bu∉I,a,b∈ZI∗(R). By Lemma 2.4(iii), u,v∉β(I) and hence u≠a or u≠b. Without loss of generality, one may assume that u≠b. But since au∈(Ru+I)∩(I:{v}), we find that ub∈I, a contradiction. Therefore, (I:{u}) is a prime ideal of R, as desired.

(ii)⇒(i) If uv ∈ I for all u,v∈ZI∗(R), then Γ_I(R) is complete and by Proposition 2.2(i), ΓI′(R) is complete. i.e., ΓI′(R)=ΓI(R). To complete the proof, we prove that if uv∉I. Then (Ru+I)∩(I:{v})=I and (Rv+I)∩(I:{v})=I. Since (I:{u})=(I:{v}) If u∈(I:{u}), then u∈(I:{v}) and hence uv∈I, a contradiction. Thus u∉(I:{u}). Also, if (Ru+I)∩(I:{u})≠I, then there exists r∈R such that ru∉I and ru2∈I. Since u2∉(I:{u}) as (I:{u}) is a prime ideal of R,r∈(I:{u}), a contradiction. Hence (Ru+I)∩(I:{u})=I. Similarly, (Rv+I)∩(I:{v})=I.

Corollary 3.4. Let I be a non radical ideal of a commutative ring R and ΓI′(R)=ΓI(R). Then the following hold.

(i) ZI(R) is an ideal of R.

(ii) β(I)2⊆I.

(iii) (I:ZI(R))=β(I).

Proof. Assume that I is not a radical ideal of R.

(i) Since I is a non radical ideal of R,β∗(I)≠ϕ. Let u∈β∗(I). Then by Lemma 2.4(iii) u is adjacent to every other vertex of ΓI′(R). Since ΓI′(R)=ΓI(R),u is adjacent to every other vertex of ΓI(R), and hence by [13, Theorem 2.5(b)] u, is adjacent to every other vertex of Γ(R/(I) and by [2, Theorem 2.5], we find that Z(R/I) is an annihilator ideal, i.e., Z(R/I)=annR/I([u]). Since Z(R/I)=annR/I([u]), we find that (I:{u})=ZI(R) and thus ZI(R) is an ideal of R.

(ii) By the first part, clearly β(I)2⊆I.

(iii) By the first part, clearly (I:ZI(R))=β(I).

Corollary 3.5. Let I be a non radical ideal of a commutative ring R. Then ΓI′(R)=ΓI(R)=Kp∨Kq¯ if and only if (I:ZI(R)) is a prime ideal.

Proof. First assume that ΓI(R)=ΓI′(R)=Kp∨Kq¯. Hence every vertex of Kp is adjacent to all the other vertices. But there is no adjacency between any two vertices of Kq¯. This implies that (I:ZI(R))=V(Kp)∪I, thus uv∉I, for every u,v∈V(Kq¯), and hence (I:{u})=(I:{v})=(I:ZI(R)). By Theorem 3.3 (I:ZI(R)) is a prime ideal of R.

Conversely since (I:ZI(R)) is a prime ideal of R, we find that uv∈I, for all u,v∈(I:ZI(R)) and uv∉I for all u,v∈ZI(R)∖(I:ZI(R)). Now it is enough to show that ΓI(R)[(I:ZI∗(R))] is complete, ΓI(R)[ZI(R) ∖(I:ZI(R))] is null graph and ΓI(R)=ΓI(R)[(I:ZI∗(R))]∨ΓI(R)[ZI(R)∖(I:ZI(R))]. We finally show that ΓI(R)=ΓI′(R). Obviously, uv∉I if and only if u,v∈ZI(R)∖(I:ZI(R)). This together with (I:ZI(R)) is a prime ideal, imply that if uv∉I, then (I:{u})=(I:{v})=(I:ZI(R)). Thus (I:{u}) is a prime ideal of R. Now by Theorem 3.3, ΓI(R)=ΓI′(R).

Corollary 3.6. Let I be a non trivial non-radical ideal of a commutative ring R. Then the following statements are equivalent.

(i) ΓI′(R) is a star graph.

(ii) gr(ΓI′(R))=∞.

(iii) ΓI(R)=ΓI′(R) and gr(ΓI(R))=∞.

(iv) (I:ZI(R)) is a prime ideal of R,|I|=|β∗(I)|=|ZI∗(R)|=2.

(v) ΓI′(R)=K1,1.

(vi) ΓI(R)=K1,1.

Proof. (i)⇒(ii) It is clear.

(ii)⇒(iii) If a∈β∗(I), then a is adjacent to every other vertex in ΓI′(R). Since gr(ΓI′(R))=∞ and ΓI(R) is a connected subgraph of ΓI′(R), we conclude that ΓI′(R)=ΓI(R), and hence gr(ΓI(R))=∞.

(iii)⇒(iv) Since I is a non trivial non radical ideal of R, it can be easily seen that ΓI′(R) is a star graph and ΓI′(R)=ΓI(R). Therefore by Corollary 3.5, (I:ZI(R)) is a prime ideal of R. Since I is a nontrivial non radical ideal of R,|I|≥2 and |β(I)|≥4. If |I|=m>2, then β(I)|=n≥6 and we can assume that u,v,w∈β∗(I) such that by Lemma 2.4(iii), u-v-w-u is a triangle and ΓI′(R) is not a star graph. Thus |I|=2. If |I|=2, then |β(I)|=4, otherwise by Lemma 2.4(iii), ΓI′(R) is not a star graph. Thus |I|=|β∗(I)|=2. If |ZI∗(R)|≥3, the we can assume that β1,β2∈β∗(I) and z∈ZI∗(R)∖β∗(I) such that by Lemma 2.4(iii), β1−β2−z−β1 forms a triangle. Hence |I|=|β∗(I)|=|ZI∗(R)|=2.

(iv)⇒(v) It is clear by Corollary 3.5.

(v)⇒(vi) It is clear.

(vi)⇒(i) It is clear.

In this section, we study the graph theoretical relationship between ΓI′(R) and Γ′(R/I) under certain parameters like clique number, max (or min) degree, vertex chromatic number, also determine a necessary and sufficient condition for ΓI′(R) to be regular and Eulerian.

Theorem 4.1. Let I be an ideal of a commutative ring R and let u,v∈ZI∗(R). Then

(i) if [u] is adjacent to [v] in Γ′(R/I), then u is adjacent to v in ΓI′(R),

(ii) if u is adjacent to v in ΓI′(R) and [u]≠[v], then [u] is adjacent to [v] in Γ′(R/I),

(iii) if u adjacent to v in ΓI′(R) and [u]=[v], then there exists r∈ZI∗(R) such that ru∉I and rv∉I, but ru2∈I and rv2∈I,

(iv) if u is adjacent to v in ΓI′(R), then all (distinct) elements of [u] and [v] are adjacent in ΓI′(R). If there exists r∈R such that ru∉I and ru2∉I, then all the distinct elements of [u] are adjacent in ΓI′(R).

Proof. (i) If [u] is adjacent to [v] in Γ′(R/I), then either (R/I)[u]∩annR/I([v])≠{I} or (R/I)[v]∩annR/I([u])≠{I}. This implies that either (Ru+I)∩(I:{v})≠I or (Rv+I)∩(I:{u})≠I. By definition u is adjacent to v in ΓI′(R).

(ii) If u is adjacent to v in ΓI′(R) then either (Ru+I)∩(I:{v})≠I or (Rv+I)∩(I:{u})≠I. Since [u]≠[v], either (R/I)[u]∩annR/I([v])≠{I} or (R/I)[v]∩annR/I([u])≠{I}. By definition [u] is adjacent to [v] in Γ′(R/I).

(iii) If u is adjacent to v in ΓI′(R), then either (Ru+I)∩(I:{v})≠I or (Ru+I)∩(I:{v})≠I. i.e., either (Ru+I)∩(I:{v})∖I≠ϕ or (Ru+I)∩(I:{v})∖I≠ϕ. Suppose that (Ru+I)∩(I:{v})∖I≠ϕ. Then there exists α∈(Ru+I)∩(I:{v})∖I such that α=ru+i for some r∈R∖I,i∈I. Clearly ruv∈I. Since [u]=[v],u=v+j for some j∈I, we find that ru2=ruu=ru(v+j)=ruv+ruj∈I. Similarly rv2∈I. Now if (Ru+I)∩(I:{v})∖I≠ϕ, then by the similar proof there exists r′∈R∖I such that r′u2,r′v2∈I.

(iv) If u is adjacent to v in ΓI′(R), then either (Ru+I)∩(I:{v})≠I or (Rv+I)∩(I:{u})≠I. Let u+i∈[u], v+j∈[v]. Then (R(u+i)+I)∩(I:{v+j})≠I or (R(v+j)+I)∩(I:{u+i})≠I. By definition u+i is adjacent to v+j in ΓI′(R).

Proposition 4.2. Let I be an ideal of a ring R. Then ΓI′(R) contains |I| disjoint subgraphs isomorphic to Γ′(R/I).

Proof. Let {aλ | λ∈Λ}⊆ZI∗(R) be a set of coset representative vertices of Γ′(R/I),i.e., V(Γ′(R/I))={[aλ]:λ∈Λ} and for each α∈I, define a graph Gα=(Vα,Eα) with Vα={aλ+α:λ∈Λ}, where aγ+α is adjacent to aδ+α in Gα whenever, [aγ] is adjacent to [aδ] in Γ′(R/I). i.e., either (R/I)[aγ]∩ann(R/I)([aδ])≠{I} or (R/I)[aδ]∩ann(R/I)([aγ])≠{I}. By Theorem 4.1 Gα is a subgraph of ΓI′(R). Also each Gα≃ΓI′(R/I), and Gα∩Gβ are disjoint if α≠β because if α≠β then V(Gα)∩V(Gβ)=ϕ.

There is a strong relation between ΓI′(R) and Γ′(R/I). Next theorem shows that how one can construct ΓI′(R) from ΓI′(R/I).

Theorem 4.3. Let ΓI′(R) be an ideal based extended zero-divisor graph of a ring R. Then we can always construct ΓI′(R) from Γ′(R/I).

Proof. Let {[aλ] | λ∈Λ} be a set of coset representative vertices of Γ′(R/I), i.e., V(Γ′(R/I))={[aλ]:λ∈Λ} and for each α∈I, define a graph Gα=(Vα,Eα) with Vα={aλ+α:λ∈Λ}, where aγ+α is adjacent to aδ+α in Gα whenever, [aγ] is adjacent to [aδ] in Γ′(R/I), i.e., either (R/I)[aγ]∩ann(R/I)([aδ])≠{I} or (R/I)[aδ]∩ann(R/I)([aγ])≠{I}. Define a graph H=(V(H),E(H)) where V(H)=∪ α∈IV(Gα) and E(H) is:

(i) all edge contained in Gα for each α∈I.

(ii) For distinct γ,δ∈Λ and for any α,β∈I,aγ+α is adjacent to aδ+β if and only if [aγ] is adjacent to [aδ] in (Γ′(R/I)).

(iii) For γ∈Λ and distinct α,β∈I,aγ+α is adjacent to aγ+β if and only if there exists a r∈R such that raγ∉I, but raγ2∈I.

Clearly, V(H)⊆V(ΓI′(R)). Note that if u∈V(ΓI′(R)), then by Theorem 4.1 [u]∈V(Γ′(R/I)) and therefore, V(ΓI′(R))⊆V(H)). So V(H)=V(ΓI′(R)). By Theorem 4.1, all edges which are defined above by (i) and (ii) are also edges in ΓI′(R). If aγ+α is adjacent to aγ+β for distinct α,β∈I, then there exists r∈R such that raγ∉I, but raγ2∈I. Therefore, (R(aγ+β)+I)∩(I:{aγ+α})≠I and (R(aγ+γ)+I)∩(I:{aγ+β)}≠I. Thus, the edges which are defined above by (iii) are also edge of ΓI′(R). Let u and v be distinct adjacent vertices of ΓI′(R). Then there exist α,β∈I and γ,δ∈Λ such that u=aγ+α and v=aδ+β. If γ≠δ and u adjacent to v in ΓI′(R). Hence by Theorem 4.1, [aγ] is adjacent to [aδ] in Γ′(R/I). Hence, the edge u-v corresponds to an edge of type (i) or (ii) of H. If γ=δ, then there exists r∈R such that raγ∉I, but raγ2∈I and the edge u-v corresponds to an edge of type (iii) of H.

Proposition 4.4. Let I be an ideal of a ring R. If Γ′(R/I) is infinite, then ΓI′(R) is infinite. If Γ′(R/I) is a graph with n vertices, then ΓI′(R) is a graph with n|I| vertices.

Proof. This is immediate from Theorem 4.3.

Definition 4.5. Let {[aλ] | λ∈Λ} be a set of coset representative vertices of Γ′(R/I). [aλ] is said to be a row of ΓI′(R), and if there exists r∈R such that raλ∉I and raλ2∈I, then we call [aλ] connected row of ΓI′(R) and ξn denote the n connected row which is contained in a maximal complete subgraph of Γ′(R/I).

Remark 4.6. Let I be an ideal in a commutative ring {R} with unity. Then every connected column of ΓI(R) defined in [13] is a connected row of ΓI′(R). By Example 2.12 and Figures 2.2 and 2.4 we observe that [2]={2,10,18} is a connected row of ΓI′(R) which is not a connected column of ΓI(R).

Figure 4. .

Theorem 4.7. Let I be a ideal in a commutative ring R. Then ω(ΓI′(R))= ξn|I|+ω(Γ′(R/I))−n.

Proof. Suppose that ω(Γ′(R/I))=k and A={[a1],[a2],⋯,[ak]}⊆V(Γ′(R/I)) such that Γ′(R/I)[A] is an induced maximal complete subgraph of Γ′(R/I). Let B=∪[ai] where [ai] is a connected row and [ai]∈A, C={ai | [ai]is a non-connected row,[ai]∈A}. Then by Theorem 4.2, ΓI′(R)[B∪C] is a complete subgraph in ΓI′(R). If B∪C∪{u} is a complete subgraph in ΓI′(R), then u∪A forms a clique of size k+1, a contradiction. Thus ΓI′(R)[B∪C] is a maximal complete subgraph. Consequently, ω(ΓI′(R))=|B∪C|=ξn|I|+ω(Γ′(R/I))−n.

Corollary 4.9. Let I be an ideal of a commutative ring R such that ΓI′(R) has no connected row. Then

(i) ω(ΓI′(R))=ω(Γ′(R/I)),

(ii) χ(ΓI′(R))=χ(Γ′(R/I)).

Proof (i) Clearly, we observe that ω(Γ′(R/I))≤ω(ΓI′(R)). Consider the case, when ω(Γ′(R/I))=k<∞, and suppose that H is a complete subgraph of ΓI′(R) with the set of (distinct) vertices u1,u2,⋯,uk+1. Since {H} is complete, we get a complete subgraph of ΓI′(R) with the set of vertices [u1],[u2],⋯,[uk+1]. Now ω(Γ′(R/I))=k implies that [ul]=[um] for some l≠m and hence ul=um+i for some i∈ I. Since H is complete, ul adjacent to um in ΓI′(R). Then we get r∈R such that ral∉I, but ral2∈I and [ul] is a connected row ΓI′(R), a contradiction. Hence ω(ΓI′(R))=k.

(ii) By Corollary 4.2, Γ′(R/I) is isomorphic to a subgraph of ΓI′(R) and hence χ(Γ′(R/I))≤χ(ΓI′(R).

Suppose that χ(Γ′(R/I))=n and C1,C2,⋯,Cn are distinct color classes of Γ′(R/I). Consider the set Sj=∪ [a]∈C j [a]. Since ΓI′(R) has no connected row, each Sj is an independent set of ΓI′(R) and V(ΓI′(R))=∪ j=1nSj.

Thus S1,S2,⋯,Sn are distinct color classes for ΓI′(R) and the graph ΓI′(R) colored by n distinct proper colors, and therefore χ(ΓI′(R)≤n. Hence χ(Γ′(R/I))=χ(ΓI′(R).

Corollary 4.9. Let I be a radical ideal of a commutative ring R . Then

(i) ω(ΓI′(R))=ω(Γ′(R/I)).

(ii) χ(ΓI′(R))=χ(Γ′(R/I)).

Theorem 4.10. Let I be an ideal in a commutative ring R. If ω(Γ′(R/I)) =χ(Γ′(R/I)), then ω(ΓI′(R))=χ(ΓI′(R)).

Proof. Suppose that ω(Γ′(R/I)) =χ(Γ′(R/I))=n. Let {aλ | λ∈Λ}⊆ZI∗(R) be a set of coset representative vertices of Γ′(R/I), i.e., V(Γ′(R/I))={[aλ]:λ∈Λ} and C1,C2,⋯,Cn are distinct color classes of Γ′(R/I). Since ω(Γ′(R/I))=n, there exists [a1],[a2],⋯,[an]∈V(Γ′(R/I)) such that any two of them lies in distinct color classes. Without loss of generality, assume that [aj]∈Cj, for all j∈{1,2⋯,n}.A={[a1],[a2],⋯,[an]}. Then Γ′(R/I)[A] is a maximal complete subgraph of Γ′(R/I). Let B={aj | [aj]∈A}∪{aj+i | [aj]∈A,raj∉I and raj2∈I  for some r∈R,i∈I∗}. Since Γ′(R/I)[A] is a maximal complete subgraph of Γ′(R/I),ΓI′(R)[B] is a maximal complete subgraph of ΓI′(R), and therefore |B|≤ω(ΓI′(R)). Hence we color the vertices of ΓI′(R) with |B| distinct colours. Clearly [a], an induced independent set of ΓI′(R) when there does not exists any r∈R such that ra∉I and ra2∈I with [a]∈A and color the vertices a+i∈[a] with the colour of a for all i∈I. Let U={a : [a]∈A}. Then U have distinct colors. For each y∉U,[y]=[at] such that t∉{1,2,⋯,n}. Since [at]∈Sj and Sj′s are independent, for each i∈I color the vertices at+i with the color of aj+i. Hence color the vertices of C=V(ΓI(R))∖U in this way, and this coloring is proper, therefore χ(ΓI(R))≤|B|. Since ω(ΓI(R))≤χ(ΓI(R)), χ(ΓI(R))=ω(ΓI(R)). This completes the proof.

Lemma 4.11. Let I be an ideal of a ring R and a∈V(ΓI′(R)). Then

deg(a)=|I|degΓ′([a]), if [a] is a non−connected row,|I|degΓ′([a])+|I|−1, if [a] is a connected row.

Proof. Clearly, deg(a)≥|I|degΓ′([a]). If [a] is connected row, then ΓI′(R)[[a]] is a complete subgraph of ΓI′(R). Thus deg(a)=|I|degΓ′([a])+|I|−1. If [a] is non-connected row, then deg(a)=|I|degΓ′([a]).

Lemma 4.12. Let I be an ideal of a ring R Then

δ(ΓI′(R))=|I|δ(Γ′(R/I))+|I|−1,if each [a]∈V(δ(Γ′(R/I)) is a connected row,|I|δ(Γ′(R/I)), otherwise.

Proof. If [a]∈V(δ(Γ′(R/I)) is a connected row, then deg(a)≤deg(b) for all b∈V(ΓI′(R) and by Lemma 4.11, deg(a)=|I|degΓ′([a])+|I|−1 ( or deg(a)=|I|δ(Γ′(R/I))+|I|−1). Thus δ(ΓI′(R))=|I|δ(Γ′(R/I))+|I|−1. Otherwise, deg(a)≤deg(b) for all b∈V(ΓI′(R) and by Lemma 4.11, deg(a)=|I|degΓ′([a]) (or deg(a)=|I|δ(Γ′(R/I)). Thus δ(ΓI′(R))=|I|δ(Γ′(R/I)).

Lemma 4.13. Let I be an ideal of a ring R Then

Δ(ΓI′(R))=|I|Δ(Γ′(R/I))+|I|−1, if each[a]∈V(Δ(Γ′(R/I)) is a non connected row,|I|Δ(Γ′(R/I)), otherwise.

Proof. If [a]∈V(Δ(Γ′(R/I)) is a non-connected row, then deg(b)≤deg(a) for all b∈V(ΓI′(R) and by Lemma 4.11, deg(a)=|I|degΓ′([a]) ( or deg(a)=|I|Δ(Γ′(R/I))). Thus Δ(ΓI′(R))=|I|δ(Γ′(R/I)). Otherwise, deg(b)≤deg(a) for all b∈V(ΓI′(R) by Lemma 4.11, deg(u)=|I|degΓ′([a]+|I|−1) (or deg(a)=|I|Δ(Γ′(R/I)+|I|−1). Thus Δ(ΓI′(R))=|I|Δ(Γ′(R/I))+|I|−1.

Theorem 4.14. Let I be an ideal in a commutative ring R. If ΓI′(R) has no connected row, then ΓI′(R) is Eulerian if and only if |I| is even or Γ′(R/I) is Eulerian.

Proof. Suppose that ΓI′(R) is Eulerian. Then deg(a) is even for all a∈V(ΓI′(R)). Since ΓI′(R) has no connected row, deg(a)=|I|degΓ′([a]) is even for all [a]∈V(Γ′(R/I)). Hence either |I| is even or degΓ′([a]) is even for all [a]∈V(Γ′(R/I)), i.e., Γ′(R/I) is Eulerian.

Conversely, assume that Γ′(R/I) is Eulerian. Hence degΓ′([a]) is even for all [a] ∈V(Γ′(R/I)). Since ΓI′(R) has no connected row, deg(a)=|I|degΓ′([a]) is even for all a ∈V(ΓI′(R). i.e., ΓI′(R) is Eulerian. If |I| is even, then ΓI′(R) is Eulerian.

Theorem 4.15. Let I be an ideal in a commutative ring R. If ΓI′(R) has a connected row, then ΓI′(R) is Eulerian if and only if |I| is odd and Γ′(R/I) is Eulerian.

Proof. Suppose that ΓI′(R) is Eulerian. Since ΓI′(R) has a connected row, there exists x∈V(ΓI′(R)) such that [x] is a connected row in ΓI′(R) and by Lemma4.11, deg(x)=|I|degΓ′[x]+|I|−1 is even. Thus we have the following cases:

Case(a) |I|degΓ′[x] and |I|−1 are odd. Then |I| is even. Since |I|degΓ′[x] is odd and |I| is even. Since |I| is even, |I|degΓ′[x] can not be odd, and this case is not possible.

Case(b) |I|degΓ′[x] and |I|−1 are even. Thus |I|degΓ′[x] is even for all [x]∈V(Γ′(R/I)). i.e., degΓ′[x] is even for all [x]∈V(Γ′(R/I)). Therefore Γ′(R/I) is Eulerian and |I| is odd.

Conversely, assume that Γ′(R/I) is Eulerian, |I| is odd and x∈V(ΓI′(R)). If [x] is a connected row, then deg(x)=|I|degΓ′[x]+|I|−1 is even and if [x] is a non-connected row, then deg(x)=|I|degΓ′[x] is also even. Hence ΓI′(R) is Eulerian.

Theorem 4.16. Let I be an ideal in a commutative ring R. If ΓI′(R) has no connected row. Then ΓI′(R) is regular if and only if Γ′(R/I) is regular.

Proof. Suppose that ΓI′(R) is regular graph, deg(x)=n for all x∈V(ΓI′(R)). Since ΓI′(R) has no connected row, by Lemma 4.11, deg(x)=|I|degΓ′[x]=n for all [x]∈V(Γ′(R/I)). Therefore degΓ′[x]=n/|I| for all [x]∈V(Γ′(R/I)). Clearly, if n is prime, then Γ′(R/I)≊K2. Otherwise Γ′(R/I) is a n|I|-regular.

Conversely, suppose that Γ′(R/I) is a regular graph. Then degΓ′[x]=n ∀ [x]∈V(Γ′(R/I)). Since ΓI′(R) has no connected row, by Lemma 4.11, for all x ∈V(Γ′′(R))deg(x)=|I|degΓ′[x]=n|I|. Therefore ΓI′(R) is n|I|-regular.

Theorem 4.17. Let I be an ideal in a commutative ring R and each row is connected. Then ΓI′(R) is n-regular, where n≠|I|−1 if and only if Γ′(R/I) is regular.

Proof. Assume that ΓI′(R) is a n-regular graph. Then deg(x)=n for all x∈V(ΓI′(R)). Since each row is connected, by Lemma 4.11, deg(x)=|I|degΓ′[x]+|I|−1, for all x∈V(ΓI′(R)) and hence degΓ′[x]=n−|I|+1|I| for all [x]∈V(Γ′(R/I)). Since degΓ′[x]≠0 and n≠|I|−1, Γ′(R/I) is a (n−|I|+1|I|)-regular graph.

Conversely, suppose that Γ′(R/I) is a regular graph. Then degΓ′[x]=p for all [x]∈V(Γ′(R/I). Since each row is connected, by Lemma 4.11, deg(x)=p|I|+|I|−1 for all x∈V(ΓI′(R). Thus ΓI′(R is a n-regular.

Theorem 4.18. Let I be an ideal of a ring R. Then 1≤χ(Γ′(R/I)) ≤χ(ΓI′(R))≤|I|χ(Γ′(R/I)).

Proof. Clearly, 1≤χ(Γ′(R/I)). Since Γ′(R/I) is isomorphic to a subgraph of ΓI′(R), χ(Γ′(R/I))≤χ(ΓI′(R)). Let χ(Γ′(R/I))=n, and C1,C2,⋯,Cn be distinct color classes for Γ′(R/I). Assume that each row is connected. Now for each 1≤j≤n, and i∈I define a set Dji={x+j : [x]∈Cj}. Since Cj's are independent, Dji are independent. Also ∪ 1≤j≤n(∪ i∈I)Dji=V(ΓI′(R). Thus {Dji : 1≤j≤n,i∈I} are distinct color classes for ΓI′(R). |I|n colors are required for colouring and this colouring is proper. Hence χ(ΓI′(R))≤|I|χ(Γ′(R/I)).

Proposition 4.19. Let I be a proper ideal of a commutative ring R. If ΓI′(R) has a connected row, then |I|≤ω(ΓI′(R)).

Proof. Assume that [u] is a connected row in ΓI′(R). Then there exists r∈R such that ru∉I and ru2∈I. If u1,u1∈[u], then (Ru1+I)∩(I:{u2})≠I and by definition u1 is adjacent to u2 in ΓI′(R). i.e., K|I| is a subgraph of ΓI′(R), and hence |I|≤ω(ΓI′(R)).

Corollary 4.20. Let I be a proper ideal of a commutative ring R such that |I|=∞. If ΓI′(R) has a connected row, then ω(ΓI′(R))=∞.

Corollary 4.21. Let I be a proper ideal of a commutative ring R such that |V(ΓI′(R))|≥2. If ΓI′(R) has a connected row, then |I|+1≤ω(ΓI′(R)).

Lemma 4.22. Let I be an ideal of a commutative ring R. Then gr(ΓI′(R))≤gr(Γ′(R/I)).

Proof. If gr(ΓI′(R))=∞, then our result holds. Now suppose that gr(Γ′(R/I))=k<∞. Let [a1]−[a2]−,⋯,−[ak]−[a1] be a cycle in ΓI′(R) with k distinct vertices. Then a1−a2−,⋯,−ak−a1 is also a cycle in ΓI′(R) of length k. Hence gr(ΓI′(R))≤k.