let $g=(v,e)$ be a simple graph. a set $ssubseteq v$ isindependent set of $g$,  if no two vertices of $s$ are adjacent.the  independence number $alpha(g)$ is the size of a maximumindependent set in the graph. in this paper we study and characterize the independent sets ofthe zero-divisor graph $gamma(r)$ and ideal-based zero-divisor graph $gamma_i(r)$of a commutative ring $r$.

let $r$ be a commutative ring. in this paper, by using algebraic properties of $r$, we study the hase digraph of prime ideals of $r$.

Graph structure is widely used to describe problems in different fields. Problems many areas, such as security and transportation, are among them. The can be solved using approaches similar the graph structure. independent set problem, which NP-Complete one of main theory modeling problems. implementation Independent problem with most significant possible number nodes called Maximum set. A lot ...

Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (...

In the classic Maximum Weight Independent Set problem, we are given a graph G with nonnegative weight function on its vertices, and goal is to find an independent set in of maximum possible weight. While problem NP-hard general, give polynomial-time algorithm working any P 6 -free graph, that is, has no path vertices as induced subgraph. This improves 5 graphs Lokshtanov et al. [ 15 ] quasipoly...

(Note that for circuits we implicitly assume that ∅ / ∈ C(M), just as we assume that for matroids ∅ ∈ I.) Proof: 1. follows from the definition that a circuit is a minimally dependent set, and therefore a circuit cannot contain another circuit. 2. Let X, Y ∈ C(M) where X 6= Y , and e ∈ X ∩ Y . From 1, it follows that X \ Y is non-empty; let f ∈ X \ Y . Assume on the contrary that (X ∪ Y ) − e i...

For any field F, the set of all functions f : V (G) → F whose sum on each maximal independent set is constant forms a vector space over F. In this paper, we show that the dimension can vary depending on the characteristic of the field. We also investigate the dimensions of these vector spaces and show that while some families, such as chordal graphs, have unbounded dimension, other families, su...

Recently, a new type of set, called random permutation set (RPS), is proposed by considering all the permutations elements in certain set. For measuring uncertainty RPS, entropy RPS presented. However, maximum principle has not been discussed. To address this issue, paper presents RPS. The analytical solution and its PMF condition are proven Besides, numerical examples used to illustrate entrop...