Manasi Mandal

Let [ ] be a commutative ring with unity, [ ] be the set of all ideals of [ ] and [ ] be a subset of [ ]. Cayley sum graph of ideals of [ ], denoted by [ ], is a simple undirected graph with vertex set [ ] and two distinct vertices [ ] and [ ] are adjacent ([ ]), whenever [ ] or [ ], for some ideal [ ] in [ ]. In this paper, we determine the girth, chromatic number, clique number, perfect code, and perfectness of the Cayley sum graphs [ ] and [ ] (where [ ] denotes the set of all nonzero maximal ideals of [ ]). Finally, we classify the family of rings whose Cayley sum graphs are split, co-graph, threshold, chordal and claw-free.

Let [ ] be a commutative ring with unity and [ ] be the set of all nontrivial proper ideals of [ ]. The intersection graph of ideals of a ring [ ], denoted by [ ], is an undirected simple graph (no loops and no parallel edges) with vertex set [ ] and two distinct vertices [ ] and [ ] are adjacent if and only if [ ] [ ] 0. In this paper, we try to establish some connections between ring theory and graph theory by studying the thickness of the line graph of the intersection graph of ideals of some rings. Here we obtain some bounds for the thickness, outerthickness and genus of the line graph of [ ], [ ]. We also study some results on the thickness and outerthickness of [ ]. Finally we determine the values of [ ] for which the graph [ ] is Eulerian and Hamiltonian.

In this paper, we first discuss about the basic semiring properties; subsemirings, ideals, [ ]-ideals, [ ]-ideals, units, structure preserving properties of both extended tropical semiring [ ] and semiring of tropical matrices [ ]. Using the known fact that the subsemimodules of [ ]-semimodule [ ] and the ideals of [ ] are in a canonical correspondence, we study different types of subsemimodules of [ ]-semimodule [ ], left (right) ideals, left (right) [ ]-ideals, left (right) [ ]-ideals of [ ]. We prove that both [ ] and [ ] are local semirings, Noetherian (Artinian), [ ]-Noetherian ([ ]-Artinian), [ ]-Noetherian ([ ]-Artinian). Finally, we deduce that [ ] is neither left nor right Noetherian (Artinian) but left as well as right [ ]-Noetherian ([ ]-Artinian).

In this article we recall a remarkable result stated as "For a fixed α, 0 < α ≤ 1, the set of all bounded statistically convergent sequences of order α is a closed linear subspace of m (m is the set of all bounded real sequences endowed with the sup norm)" by Bhunia et al. (Acta Math. Hungar. 130 (1-2) (2012), 153-161) and to develop the objective of this perception we demonstrate that the set of all bounded statistically convergent sequences of order α may not form a closed subspace in other sequence spaces. Also we determine two different sequence spaces in which the set of all statistically convergent sequences of order α (irrespective of boundedness) forms a closed set.