Complex non-backtracking matrix for directed graphs

In network analysis, various matrix representations have been developed to investigate the structural properties of the corresponding network, among which the non-backtracking (NBT) matrix is one such representation. The NBT matrix is well-known for its relationship with the Ihara zeta function [15] defined as an infinite product over equivalence classes of primitive cycles. It was shown in [15, 36] that, for regular graphs, the reciprocal of the Ihara zeta function can be expressed as a polynomial related to the adjacency matrix. The relation between the zeta function and the determinant of the NBT matrix was elucidated in [12], extended to irregular graphs in [3], and an elementary proof was provided in [35]. The connection between the polynomial and the determinant of the NBT matrix, via the Ihara zeta function, is known as the Ihara's formula.