Abstract
Graph invariants are the properties of graphs that do not change under graph isomorphisms, the independent set decision problem, vertex covering problem, and matching number problem are known to be NP-Hard, and hence it is not believed that there are efficient algorithms for solving them. In this paper, the graph invariants matching number, vertex covering number, and independence number for the zero-divisor graph over the rings Z p k and Z p k q r are determined in terms of the sets S p i and S p i q j respectively. Accordingly, a formula in terms of p , q , k , and r, with n = p k , n = p k q r is provided.
Highlights
The independent set decision problem, vertex covering problem, and matching number problem are known to be classical optimization problems in computer science and are typical examples of NP-Hard problems [1,2,3], and it is not believed that there are efficient algorithms for solving them but rather have approximation algorithms
Beck was mainly interested in graph coloring. He defined the vertices of the graph as the elements of a commutative ring R and two different vertices x and y are adjacent if xy = 0
We find the matching number, vertex covering number, and the independence number of zero-divisor graphs of Zn, where the set V (Γ( Zn )) = { x : x ∈ Zn∗ and gcd( x, n) 6= 1}
Summary
The independent set decision problem, vertex covering problem, and matching number problem are known to be classical optimization problems in computer science and are typical examples of NP-Hard problems [1,2,3], and it is not believed that there are efficient algorithms for solving them but rather have approximation algorithms. The graph invariants matching number, vertex covering number, and independence number for the zero-divisor graph over the rings Z pk and. Anderson and Livingston introduced the zero-divisor graph of a commutative ring R [5]. They defined the set of vertices of the graph to be the nonzero zero-divisors of R and two different vertices x and y are adjacent if xy = 0. We find the matching number, vertex covering number, and the independence number of zero-divisor graphs of Zn , where the set V (Γ( Zn )) = { x : x ∈ Zn∗ and gcd( x, n) 6= 1}. The independence number of G is the maximum size of an independent set in G and is denoted by α( G )
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