On the relation between the adjacency rank of a complex unit gain graph and the matching number of its underlying graph

摘要

Let G(phi) be an n-vertex complex unit gain graph and let G be its underlying graph. The adjacency rank of G(phi), written as r(G(phi)), is the rank of its adjacency matrix and denote by alpha'(G) the matching number of the underlying graph G. In this contribution, based on combinatorial interpretation of all the coefficients of the characteristic polynomial of G(phi), we determine sharp upper and lower bounds on r(G(phi)) - 2 alpha'(G). Furthermore, we establish sharp lower bounds on r(G(phi)) -alpha'(G) and r(G(phi)) )/alpha'(G). All the corresponding extremal complex unit gain graphs are characterized.

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