Refined Lower Bounds for the Laplacian Estrada Index of Connected Graphs via the Two Largest Degrees

Let G be a graph with n vertices and Laplacian eigenvalues μ 1 , μ 2 , …, μ n . The Laplacian Estrada index of G is defined as . In this paper, using the Karush–Kuhn–Tucker optimization framework under inequality constraints, we establish new lower bounds for LEE( G ) in terms of the two largest degrees of G . Specifically, if G is a connected graph with n ≥ 3 vertices, m edges, and degree sequence d 1 ≥ d 2 ≥ ⋯≥ d n , then if d 3 + ⋯+ d n ≤ ( n − 3) d 2 and otherwise. These inequalities improve previously known results on the Laplacian Estrada index.