## 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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### References:

 [1] DOI: 10.1080/03081088508817681 · Zbl 0594.05046 [2] DOI: 10.1137/S0895480191222653 · Zbl 0795.05092 [3] Horn, R. A. and Johnson, C. R. 1985.Matrix Analysis, 189–190. New York: Cambridge U.P. [4] DOI: 10.1016/S0024-3795(98)10149-0 · Zbl 0931.05052 [5] DOI: 10.1016/S0024-3795(98)10148-9 · Zbl 0931.05053 [6] DOI: 10.1016/0024-3795(94)90486-3 · Zbl 0802.05053 [7] Mohar B., Graph Theory, Combinatorics and Applications pp 871– (1991)
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