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Via the process of isospectral graph reduction the adjacency matrix of agraph can be reduced to a smaller matrix while its spectrum is preserved up tosome known set. It is then possible to estimate the spectrum of the originalmatrix by considering Gershgorin-type estimates associated with the reducedmatrix. The main result of this paper is that eigenvalue estimates associatedwith Gershgorin, Brauer, Brualdi, and Varga improve as the matrix size isreduced. Moreover, given that such estimates improve with each successivereduction, it is also possible to estimate the eigenvalues of a matrix withincreasing accuracy by repeated use of this process.

Isospectral Graph Reductions and Improved Estimates of Matrices' Spectra