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A note on the second largest eigenvalue of the Laplacian matrix of a graph. (English) Zbl 0979.15016
The Laplacian matrix of a connected simple graph \(G\) is defined as \(L(G)=D(G)-A(G)\) where \(A(G)\) is the adjacency matrix of \(G\) and \(D(G)=diag(d_1(G), \dots, d_n(G))\) is a diagonal matrix whose nonzero elements are the degrees of vertices of \(G\) and \(d_1(G) \geq d_2(G) \geq \dots \geq d_n(G)\). The Laplacian matrix \(L(G)\) is a singular positive semidefinite matrix with eigenvalues \(\lambda_1(G) \geq \lambda_2(G) \geq \dots \lambda_n(G)=0\).
The authors show that the second largest eigenvalue of the Laplacian matrix \(L(G)\) can be bounded using the second largest degree of \(G\), \(\lambda_2(G) \geq d_2(G)\). This lower bound is satisfied for any connected graph with \(n \geq 3\) vertices. They mention special cases when the equality is fulfilled.

MSC:
15A42 Inequalities involving eigenvalues and eigenvectors
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C40 Connectivity
15B57 Hermitian, skew-Hermitian, and related matrices
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