Jacobi symbol cordial labeling and Jacobi symbol 3-equitable labeling of digraphs

Abstract

Let [Formula: see text] be a simple digraph and a bijection [Formula: see text]. The labeling [Formula: see text] induces on arc labeling [Formula: see text] such that for any arc [Formula: see text] in [Formula: see text], [Formula: see text], where [Formula: see text] is the Jacobi symbol. Let [Formula: see text] be the number of arcs labeled with [Formula: see text]. The labeling [Formula: see text] is called a Jacobi symbol labeling if [Formula: see text] for all [Formula: see text]. Moreover, [Formula: see text] is Jacobi symbol digraph if it admits Jacobi symbol labeling. The labeling [Formula: see text] is called a Jacobi symbol cordial labeling if [Formula: see text]. Moreover, [Formula: see text] is Jacobi symbol cordial digraph if it admits Jacobi symbol cordial labeling. The labeling [Formula: see text] is called a Jacobi symbol 3-equitable labeling if [Formula: see text] for all [Formula: see text]. A digraph which admits a Jacobi symbol 3-equitable labeling is called a Jacobi symbol 3-equitable digraph. In this paper, we will find out some Jacobi symbol digraphs, Jacobi symbol cordial digraphs and Jacobi symbol 3-equitable digraphs.