Let R be a noncommutative ring. The zero-divisor graph of R, denoted by Γ(R), is the (directed) graph with vertices Z(R)∗ = Z(R)− {0}, the set of nonzero zero-divisors of R, and for distinct x, y ∈ Z(R)∗, there is an edge x → y if and only if xy = 0. In this paper we investigate the zero-divisor graph of triangular matrix rings over commutative rings. Mathematics Subject Classification: 16S70; ...

Let $R$ be a commutative ring with identity and $mathbb{A}(R)$ be the set   of ideals of $R$ with non-zero annihilators. In this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of $R$, denoted by $mathbb{AG}_P(R)$. It is a (undirected) graph with vertices $mathbb{A}_P(R)=mathbb{A}(R)cap mathbb{P}(R)setminus {(0)}$, where   $mathbb{P}(R)$ is...

The notion of graph powers is a well-studied topic in graph theory and its applications. In this paper, we investigate a bipartite analogue of graph powers, which we call bipartite powers of bigraphs. We show that the classes of bipartite permutation graphs and interval bigraphs are closed under taking bipartite power. We also show that the problem of recognizing bipartite powers is NP-complete...

A bipartite graph G = (L,R;E) where V (G) = L∪R, |L| = p, |R| = q is called a (p, q)-tree if |E(G)| = p+ q − 1 and G has no cycles. A bipartite graph G = (L,R;E) is a subgraph of a bipartite graph H = (L′, R′;E′) if L ⊆ L′, R ⊆ R′ and E ⊆ E′. In this paper we present sufficient degree conditions for a bipartite graph to contain a (p, q)-tree.