Zero-divisor graphs of partial transformation semigroups

Let P-n be the partial transformation semigroup on X-n = {1, 2, ... , n}. In this paper, we find the left zero divisors, right zero-divisors and two sided zero-divisors of P-n, and their numbers. For n >= 3, we define an undirected graph Gamma(P-n) associated with P-n whose vertices are the two sided zero-divisors of P-n excluding the zero element theta of P-n with distinct two vertices alpha and beta joined by an edge in case alpha beta = 0 = beta alpha. First, we prove that Gamma(P-n) is a connected graph, and find the diameter, girth, domination number and the degrees of the all vertices of Gamma(P-n). Furthermore, we give lower bounds for clique number and chromatic number of Gamma(P-n).

Automated summary

The paper explores the zero divisors and their properties of the partial transformation semigroup P-n, and defines an undirected graph Gamma(P-n) associated with P-n. It investigates the connectivity, diameter, girth, domination number, and degrees of the vertices of Gamma(P-n), and provides lower bounds for the clique number and chromatic number of Gamma(P-n).