• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.359-370
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• 2020

A graph G is called a well covered graph if every maximal independent set in G is maximum, and co-well covered graph if its complement is a well covered graph. We study some properties of a co-well covered graph and we characterize when the join, the corona product, and cartesian product are co-well covered graphs. Also we characterize when powers of trees and cycles are co-well covered graphs. The line graph of a graph which is co-well covered is also studied.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.969-975
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• 2022

Let R be a finite commutative ring with nonzero unity and let Z(R) be the zero divisors of R. The total graph of R is the graph whose vertices are the elements of R and two distinct vertices x, y ∈ R are adjacent if x + y ∈ Z(R). The total graph of a ring R is denoted by 𝜏(R). The independence number of the graph 𝜏(R) was found in [11]. In this paper, we again find the independence number of 𝜏(R) but in a different way. Also, we find the independent dominating number of 𝜏(R). Finally, we examine when the graph 𝜏(R) is well-covered.