On the inclusion ideal graph of a ring

On the metric determination of linear dependence graph

“…Beck 1 was the pioneer to initiate the study on this topic when he determined a graph associated with a commutative ring by means of coloring. Later on, this topic gained the attention of many researchers which resulted a number of remarkable determinations of various types of graphs associated with groups, 2‐12 rings, 13‐15 ideals, 16‐18 vector spaces, 19‐24 and posets 25‐28 …”

“…Beck 1 was the pioneer to initiate the study on this topic when he determined a graph associated with a commutative ring by means of coloring. Later on, this topic gained the attention of many researchers which resulted a number of remarkable determinations of various types of graphs associated with groups, [2][3][4][5][6][7][8][9][10][11][12] rings, [13][14][15] ideals, [16][17][18] vector spaces, [19][20][21][22][23][24] and posets. [25][26][27][28] The problem of vertex determination in a connected graph G by defining the metric on G attracted various graph theorist as well many other researchers due to its significant applications in several extents including verification, security and discovery of networks, 29 the chemistry of pharmaceutics for drug designing, 30 mastermind game strategies, 31 navigation of robots, 32 connected joins in graphs, 33 and solution of coin weighing problems.…”

On the inclusion ideal graph of semigroups

“…The investigation of graphs related to various algebraic structures is very important because graphs of this type have valuable applications and are related to automata theory(see [13,15,16]). Akbari et al [2] have introduced the notion of inclusion ideal graph associated with ring structure. The inclusion ideal graph In(R) of a ring R is an undirected simple graph whose vertex set is the collection of nontrivial left ideals of R and two distinct nontrivial left ideals I, J are adjacent if and only if either I ⊂ J or J ⊂ I.…”

“…It is pertinent as well as interesting to associate graphs to ideals of a semigroup as ideals gives a lot of information about structure of semigroups [23,24]. Motivated with the work of [2,8], in this paper, we consider the inclusion ideal graph associated with semigroups. The inclusion ideal graph In(S) of a semigroup S is an undirected simple graph whose vertex set is nontrivial left ideals of S and two distinct nontrivial left ideals I, J are adjacent if and only if I ⊂ J or J ⊂ I.…”