A (1, ≤ )-identifying code in a digraph D is a subset C of vertices of D such that all distinct subsets of vertices of cardinality at most have distinct closed inneighbourhoods within C. In this paper, we give some sufficient conditions for a digraph of minimum in-degree δ − ≥ 1 to admit a (1, ≤ )-identifying code for = δ − , δ − + 1. As a corollary, we obtain the result by Laihonen that states that a graph of minimum degree δ ≥ 2 and girth at least 7 admits a (1, ≤ δ)-identifying code. Moreover, we prove that every 1-in-regular digraph has a (1, ≤ 2)-identifying code if and only if the girth of the digraph is at least 5. We also characterize all the 2-in-regular digraphs admitting a (1, ≤ )-identifying code for = 2, 3.Mathematics Subject Classifications: 05C69, 05C20

A (1, ≤ )-identifying code in digraph D is a dominating subset C of vertices of D, such that all distinct subsets of vertices of D with cardinality at most have distinct closed in-neighborhoods within C. As far as we know, it is the very first time that the spectral graph theory has been applied to the identifying codes. We give a new method to obtain an upper bound on for digraphs. The results obtained here can also be applied to graphs.

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