Signed Graphs with extremal least Laplacian eigenvalue

A signed graph is a pair Γ = (G, σ), where G = (V (G), E(G))is a graph and σ : E(G) → {+, −} is the corresponding sign function. For a signed graph we consider the Laplacian matrix defined as L(Γ) = D(G) − A(Γ), where D(G) is the matrix of vertex degrees of G and A(Γ) is the (signed) adjacency matrix. It is well-known that Γ is balanced, that is, each cycle contains an even number of negative edges, if and only if the least Laplacian eigenvalue λn = 0. Therefore, if Γ is not balanced, then λn > 0. We show here that among unbalanced connected signed graphs of given order the least eigenvalue is minimal for an unbalanced triangle with a hanging path, while the least eigenvalue is maximal for the complete graph with the all-negative sign function.