### Some Reduction Formulas and the Characterization of Singular and Nonsingular Directed Fans

Severino V. Gervacio | Isagani B. Jos

**Discipline: **Mathematics

#### Abstract:

A digraph is called singular or nonsingular according as its adjacency matrix is singular or nonsingular. An expression of the determinant of the adjacency matrix of a digraph in terms of the determinant of smaller digraphs obtained from the given one is called a reduction formula. Reduction formulas are established in this paper. Furthermore, using these reduction formulas, we determine which of the directed fans are singular. Moreover, we show that if a directed fan Fn is nonsingular, then the determinant of its adjacency matrix is (-1)^{n}.

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