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Keywords:
adjacency matrix; Laplacian matrix; normalized adjacency matrix; spectral radius; algebraic connectivity; Randić index
adjacency matrix; Laplacian matrix; normalized adjacency matrix; spectral radius; algebraic connectivity; Randić index
Summary:
For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first, we derive bounds for the second largest and the smallest eigenvalues of adjacency matrices of $k$-regular graphs. Then we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest and the largest eigenvalues of the Laplacian matrices of graphs. The sharpness of these bounds is verified by examples.
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