Upper bounds of spectral radius of symmetric matrices and graphs

Abstract

The spectral radius ρ(A) is the maximum absolute value of the eigenvalues of a matrix A. In this paper, we establish some relationship between the spectral radius of a symmetric matrix and its principal submatrices, i.e., if A is partitioned as a 2×2 block matrix A=(0A12A21A22), then ρ(A)2≤ρ22+θ⁎, where θ⁎ is the largest root of the equation μ2=(x−ν)2(ρ22+x) and ρ2=ρ(A22), μ=ρ(A12A22A21), ν=ρ(A12A21). Furthermore, the results are used to obtain several upper bounds of the spectral radius of graphs, which strengthen or improve some known results.