On Graphs of Bounded Semilattices
P. Malakooti Rad, Peyman Nasehpour · Mathematical Notes · 2020
In this paper, we introduce the graph G ( S ) of a bounded semilattice S , which is a generalization of the intersection graph of the substructures of an algebraic structure. We prove some general theorems about these graphs; as an example, we show that if S is a product of three or more chains, then G ( S ) is Eulerian if and only if either the length of every chain is even or all the chains are of length one. We also show that if G ( S ) contains a cycle, then girth( G ( S )) = 3. Finally, weshowthat if ( S , +, ·, 0, 1) is a dually atomic bounded distributive lattice whose set of dual atoms is nonempty, and the graph G ( S ) of S has no isolated vertex, then G ( S ) is connected with diam( G ( S )) ≤ 4.