EUDML

Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_{1}, d_{2}, ..., d_{n})$ . Denote ${(^{α} t)}_{i} = \sum_{j : i \sim j} d_{j}^{α}$ , ${(^{α} m)}_{i} = {(^{α} t)}_{i} / d_{i}^{α}$ and ${(^{α} N)}_{i} = \sum_{j : i \sim j} {(^{α} t)}_{j}$ , where $α$ is a real number. Denote by $λ_{1} (G)$ and $μ_{1} (G)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$ , respectively. In this paper, we present some upper and lower bounds of $λ_{1} (G)$ and $μ_{1} (G)$ in terms of ${(^{α} t)}_{i}$ , ${(^{α} m)}_{i}$ and ${(^{α} N)}_{i}$ . Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.

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Subdirect sums of doubly diagonally dominant matrices.

On symmetric matrices with exactly one positive eigenvalue.

Inequalities for the minimum eigenvalue of $M$ -matrices.

A note on estimates for the spectral radius of a nonnegative matrix.

A weak regular splitting for singular linear systems.

Regions containing eigenvalues of a matrix.

Preconditioned diagonally dominant property for linear systems with $H$ -matrices.

Some new results on determinantal inequalities and applications.

A note on generalized Perron complements of $Z$ -matrices.

Some bounds for the spectral radius of the Hadamard product of matrices.

Bounds for the (Laplacian) spectral radius of graphs with parameter $α$