The annihilators comaximal graph

In this paper, we introduce and study a new kind of graph related to a unitary module [Formula: see text] on a commutative ring [Formula: see text] with identity, namely the annihilators comaximal graph of submodules of [Formula: see text], denoted by [Formula: see text]. The (undirected) graph [Formula: see text] is with vertices of all non-trivial submodules of [Formula: see text] and two vertices [Formula: see text] of [Formula: see text] are adjacent if and only if their annihilators are comaximal ideals of [Formula: see text], i.e. [Formula: see text]. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of [Formula: see text] and the module-theoretic properties of [Formula: see text]. We study the annihilators comaximal graph [Formula: see text] in terms of the powers of the decomposition of [Formula: see text] to product distinct prime numbers in some special cases.

Comaximal graph of amalgamated algebras along an ideal

Let [Formula: see text] and [Formula: see text] be commutative rings with identity, [Formula: see text] be an ideal of [Formula: see text], and let [Formula: see text] be a ring homomorphism. The amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text] denoted by [Formula: see text] was introduced by D’Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of [Formula: see text] which are transferred to the comaximal graph of [Formula: see text], and also we study some algebraic properties of the ring [Formula: see text] by way of graph theory. The comaximal graph of [Formula: see text], [Formula: see text], was introduced by Sharma and Bhatwadekar in 1995. The vertices of [Formula: see text] are all 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]. Let [Formula: see text] be the subgraph of [Formula: see text] generated by non-unit elements, and let [Formula: see text] be the Jacobson radical of [Formula: see text]. It is shown that the diameter of the graph [Formula: see text] is equal to the diameter of the graph [Formula: see text], and the girth of the graph [Formula: see text] is equal to the girth of the graph [Formula: see text], provided some special conditions.

On the normalized Laplacian spectrum of the comaximal graphs

Let [Formula: see text] be a commutative ring with nonzero identity. The comaximal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with 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]. Let [Formula: see text] be an induced subgraph of [Formula: see text] with nonunit elements of [Formula: see text] as vertices. In this paper, we describe the normalized Laplacian spectrum of [Formula: see text], and we determine it for some values of [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text]. Moreover, we investigate the normalized Laplacian energy and general Randic index of [Formula: see text].

On the projective comaximal graphs of lattices

In this paper, we investigate the projectivity of the comaximal graph of a finite lattice.

Independence complexes of comaximal graphs of commutative rings with identity

We study topology of the independence complexes of comaximal (hyper)graphs of commutative rings with identity. We show that the independence complex of comaximal hypergraph is contractible or homotopy equivalent to a sphere, and that the independence complex of comaximal graph is almost always contractible.