On the second smallest and the largest normalized Laplacian eigenvalues of a graph. (English) Zbl 1470.05106

Tian, Xiao-guo; Wang, Li-gong; Lu, You

Acta Math. Appl. Sin., Engl. Ser. 37, No. 3, 628-644 (2021).

Summary: Let \(G\) be a simple connected graph with order \(n\). Let \(\mathcal{L}(G)\) and \(\mathcal{Q}(G)\) be the normalized Laplacian and normalized signless Laplacian matrices of \(G\), respectively. Let \(\lambda_k(G)\) be the \(k\)-th smallest normalized Laplacian eigenvalue of \(G\). Denote by \(\rho (A)\) the spectral radius of the matrix \(A\). In this paper, we study the behaviors of \(\lambda_2(G)\) and \(\rho(\mathcal{L}(G))\) when the graph is perturbed by three operations. We also study the properties of \(\rho(\mathcal{L}(G))\) and \(X\) for the connected bipartite graphs, where \(X\) is a unit eigenvector of \(\mathcal{L}(G)\) corresponding to \(\rho(\mathcal{L}(G))\). Meanwhile we characterize all the simple connected graphs with \(\rho(\mathcal{L}(G))=\rho(\mathcal{Q}(G))\).

MSC:

05C50	Graphs and linear algebra (matrices, eigenvalues, etc.)
15A18	Eigenvalues, singular values, and eigenvectors

Keywords:

second smallest normalized Laplacian eigenvalue; normalized Laplacian spectral radius; normalized signless Laplacian spectral radius

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