Bounds of nullity for complex unit gain graphs

A complex unit gain graph, or T-gain graph, is a triple Phi = ( C, T , p ) comprised of a simple graph C as the underlying graph of Phi, the set of unit complex numbers T = { z is an element of C : | z | = 1}, }, and a gain function p : (E) over right arrow -> T with the property that p ( e ij ) = p ( e ji ) -1 . A cactus graph is a connected graph in which any two cycles have at most one vertex in common. In this paper, we firstly show that there does not exist a complex unit gain graph with nullity n ( C ) -2 m ( C ) +2c(C) c ( C ) -1, where n ( C ), m ( C ) and c ( C ) are the order, matching number, and cyclomatic number of C . Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30]. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.