The Eigenvalues-Based Entropy and Spectrum of the Directed Cycles

2. Preliminary

Let be a directed graph, where is the set of vertices and is the set of ordered vertex pairs , which are edges of the directed graph. The edge is also denoted by . The basic property of a vertex in a digraph is the degree of the vertex. The in-degree and out-degree of a vertex are denoted by and , respectively. The in-degree sum of vertex is denoted by . The degree matrix is defined as the diagonal matrix, where each vertex’s degree is at the position of the main diagonal, and the term outside the main diagonal is zero. Similarly, in a digraph, the in-degree and the out-degree matrix can be defined as and , respectively. Let the in-degree adjacency matrix of the digraph be denoted by , in-degree Laplacian matrix be denoted by , and in-degree signless Laplacian matrix be denoted by (Cvetkovic et al., 1998).