Mixed graphs are graphs whose edges may be directed or undirected. The first eigenvalue of a mixed graph is the least nonzero eigenvalue of its Laplacian matrix. We determine the unique mixed graphs with minimum first eigenvalue over all nonsingular mixed unicyclic graphs with fixed number of branch vertices, and the unique graph with minimum least signless Laplacian eigenvalue over all nonbipartite unicyclic graphs with fixed number of branch vertices.
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