The relationship between Laplacian eigenvalues and edge connectivity in graphs

A simple graph G is in the set ζ3 of graphs if there exist four disjoint proper subsets X1,X2,X3,X4 of the vertex set V such that V∖(∪i=14Xi)≠φ and the edge connectivity λ(G)=e(Xi,V∖Xi) for 1≤i≤4. The eigenvalues of the Laplacian matrix L(G)=D(G)−A(G) are typically arranged in nondecreasing order: λ1G,1,−1≤λ2G,1,−1≤⋯≤λnG,1,−1. Where λi(G,1,−1) denotes the i-th largest eigenvalue of L(G,1,−1). The matrix L(G,a,b) is defined as follows: let a,b be two real numbers such that a∈[−1,+∞),b≠0 and ab∈[−1,+∞), the matrix bA(G), where A(G) and D(G) are the adjacency matrix and the degree matrix of the graph G, respectively. In this work, we investigated the relationship between the λn−4(G,1,−1) and λ(G) when G∈ζ3. Additionally, we extend some results on L(G,1,−1) to matrices L(G,a,b).