THE UNIT GRAPH OF A COMMUTATIVE SEMIRING

Let S be a commutative semiring with unity and U(S) be the set of all units of S. The unit graph of S, denoted by G(S) is the undirected graph with vertex set S and two distinct vertices x and y are adjacent in G(S) if and only if x + y ∈ U(S). In this paper, we concentrate on the unit graph G(S) and look at several properties like the completeness, the bipartiteness, the connectedness, the diameter and the girth. We also obtain necessary and sufficient conditions for G(S) to be traversable under certain conditions.