Laplacian eigenvalue conditions for edge-disjoint spanning trees and a forest with constraints

Let k be a positive integer and let G be a simple graph of order n with minimum degree δ . A graph G is said to have property P(k,d) if it contains k edge-disjoint spanning trees and an additional forest F with edge number |E(F)|>d−1d(|V(G)|−1), such that if F is not a spanning tree, then F has a component with at least d edges. Let D(G) be the degree diagonal matrix of G . We denote λ_i and μ_i as the i th largest eigenvalue of the adjacency matrix A(G) of G and the Laplacian matrix L(G)=D(G)−A(G) of G for i=1,2,…,n, respectively. In this paper, we investigate the relationship between Laplacian eigenvalues and property P(k,δ). Let t be a positive integer, and define G_t as the set of simple graphs such that each G∈G_t contains at least t+1 non-empty disjoint proper subsets V₁,V₂,…,V_t+1 satisfying V(G)∖⋃_i=1^t+1V_i≠∅ and edge connectivity κ^′(G)=e(V_i,V(G)∖V_i) for any i=1,2,…,t+1. For the class of graphs G₁ with minimum degree δ , we provide a sufficient condition involving the third smallest Laplacian eigenvalue μ_n−2(G) for a graph G∈G₁ to have property P(k,δ). Similarly, for the class of graphs G₂ with minimum degree δ , we establish a corresponding sufficient condition involving the fourth smallest Laplacian eigenvalue μ_n−3(G) for a graph G∈G₂ to have property P(k,δ). Furthermore, we extend the spectral conditions for all the results about μ_n−2(G), μ_n−3(G) and λ₂(G) to the general graph matrices aD(G)+A(G) and aD(G)+bA(G).