The multiplicity of an [formula omitted]-eigenvalue: A unified approach for mixed graphs and complex unit gain graphs

Abstract

The work of Wang et al. (2020) established an upper bound on the multiplicity of a real number as an adjacency eigenvalue of an undirected simple graph G according to the dimension of its cycle space and the number of its pendants. The work of Cardoso et al. (2018) studied the multiplicity of α as an eigenvalue of αD(G)+(1−α)A(G),α∈[0,1), where D(G) is the diagonal matrix of degrees and A(G) is the adjacency matrix of G, which was ported to signed graphs by the work of Belardo et al. (2019). Here, on the one hand, we consider both the Wang–Wei–Jin-type and Cardoso–Pastén–Rojo-type routines developed for graphs to the level of mixed graphs and complex unit gain graphs. On the other hand, we consider Belardo–Brunetti–Ciampella-type routines developed for signed graphs to the level of mixed graphs and complex unit gain graphs. We show that, with suitable adaption, such routines can be successfully ported to mixed graphs and complex unit gain graphs. By these obtained results in the current paper, the corresponding results for undirected graphs, oriented graphs and signed graphs are deduced.