The Eigenvalues-Based Entropy and Spectrum of the Directed Cycles

The directed cycles form a foundational structure within a network model. By analyzing the in-degree characteristic polynomial of three kinds of matrices of the directed cycles, the authors obtain the eigenvalues of the adjacency matrix A(-), the Laplacian matrix L-, and the signless Laplacian matrix Q(-). This study investigates the eigenvalues spectrum of these three types of matrices for directed cycles and introduces an eigenvalue-based entropy calculated from the real part of the eigenvalues. The computer simulation reveals interesting characteristics on the spectrum of the signless Laplacian. The concept of eigenvalue-based entropy holds promise for enhancing our understanding of graph neural networks and more applications of social networks.