The Laplacian spectral ratio R_L(G) of a connected graph G is defined as the ratio of its Laplacian spectral radius to its algebraic connectivity. If a connected graph of even order satisfies R_L(G)>=2, then G has a perfect matching (Brouwer and Haemers 2005, Lin et al. 2023). If Δ is the maximum vertex degree and δ is the minimum vertex degree, then for a connected graph other than a complete graph, R_L>=(Δ + 1)/δ .

algebraic connectivity | Laplacian matrix | Laplacian polynomial | Laplacian spectral radius