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Let L be a lattice with the least element 0, and I be an ideal of L. In this paper, we introduce a generalized zero-divisor graph G (L) I Γ of L with respect to the ideal I. We show that G (L) I Γ is connected with diameter at most three. If G (L) I Γ has a cycle, we show that the girth of G (L) I Γ is at most four. We also investigate the existence of cut vertices of G (L) I Γ Moreover, we examine certain situations when G (L) I Γ is a complete bipartite graph.

Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, and define several generalized zero-divisor graphs of R with respect to I. In this paper, we investigate the ring-theoretic properties of R and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals.

Let N be a near-ring and I be an ideal of N. The graph of N with respect to I is a graph with V (N ) as vertex set and any two distinct vertices x and y are adjacent if and only if xNy ⊆ I oryNx ⊆ I. This graph is denoted by GI(N). We define the line graph of GI(N) as a graph with each edge of GI (N ) as vertex and any two distinct vertices are adjacent if and only if their corresponding edges share a common vertex in the graph GI (N ). We denote this graph by L(GI (N )). We have discussed the diameter, girth, clique number, dominating set of L(GI(N)). We have also found conditions for the graph L(GI(N)) to be acycle graph.

The Unitary addition Cayley graph of Gaussian integers modulo n denoted by \(G_n[i]\) has a vertex set \(\mathbb {Z}_n[i]\). Any two vertices, \(x=a_1+ib_1\), \(y=a_2+ib_2\) of \(G_n[i]\), are adjacent if and only if \(gcd(N(x+y),n)=1\), where \(N(a+ib)=a^2+b^2\) is the norm of any element \(a+ib\) in \(\mathbb {Z}_n[i]\). First, we find two important results of unitary addition Cayley graph of Gaussian integers modulo n by considering \(\mathbb {Z}_n[i]\) and \(U_n[i]\) as the vertex set. Using the results, we obtain clique number, chromatic number of the graph \(G_n[i]\). Lastly, domination number of unitary addition Cayley graph of Gaussian integers modulo n is also found in this paper.

Millions of people use online social networking sites such as Facebook, Twitter, WhatsApp, Instagram etc. Nowadays, social media is very popular among all of us especially young generation and has become a vital part of our life. Just like a graph is made up of vertices and edges , a social media network is made up of persons or communities where each person or community represents the vertex of the social graph and the adjacency of the vertices ( edges ) is determined via friendship, common interest, common liking etc. In this paper we try to show the use of some graphical parameter in social graph so that the many properties of social media network can be reflected through graph. We have also shown how domination number plays an important role in social graph.

Let R be a commutative ring with identity and M be an R-module. Let Z(R) be the set of zero-divisors of R and Z(M) be the set of annihilators over M in R. The total graph of idealization, \(T\left( \Gamma (R(+)M)\right) \) was defined as the graph with all elements of the ring \(R(+)M =\lbrace (r,m)\mid r \in R, m \in M\rbrace \) as vertices where any two distinct vertices \((x,m), (y,n) \in R(+)M\) are adjacent if and only if \((x,m) + (y,n) \in Z(R(+)M)\), the set of zero-divisors of \(R(+)M\). In this paper we define the line graph of total graph of idealization, denoted by \(L\left( T(\Gamma (R(+)M))\right) \) as the graph with all the edges of \(T\left( \Gamma (R(+)M)\right) \) as vertices and any two distinct vertices are adjacent if and only if their corresponding edges share a common vertex in the graph \(T\left( \Gamma (R(+)M)\right) \). In this paper we discuss the diameter, girth and clique number of the graph \(L\left( T(\Gamma (R(+)M))\right) \). We also find condition for the graph \(L\left( T(\Gamma (R(+)M))\right) \) to be connected when \(Z(R(+)M)\) is not an ideal of the \(R(+)M\).

Let R be a commutative ring with unity. Let Z(R) be the set of all zero-divisors of R. For x  Z(R), let

Let be a commutative ring. The prime graph of the ring is defined as a graph whose vertex set consists of all elements of and any two distinct vertices x and y are adjacent if and only if or . This graph is denoted by . In this paper we investigate some relations between the chromatic number of prime graph of finite product of commutative rings and the chromatic number of prime graph of these rings. We also obtain some results on the chromatic number of prime graph of the ring .

The main purpose of this paper is to study the zero-divisor graph for direct product of finite commutative rings. In our present investigation we discuss the zero-divisor graphs for the following direct products: direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and multiplication modulo p for a prime number p, direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and multiplication modulo 2p for an odd prime number p and direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and multiplication modulo p – 2 for that odd prime p for which p – 2 is a prime number. The aim of this paper is to give some new ideas about the neighborhood, the neighborhood number and the adjacency matrix corresponding to zero-divisor graphs for the above mentioned direct products. Finally, we prove some results of annihilators on zerodivisor graph for direct product of A and B for any two commutative rings A and B with unity

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