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The maximum multiplicity among eigenvaluesof matriceswith a given graph cannot generally be expressed in terms of the degrees of the vertices (even when the graph is a tree). Given are best possible lower and upper bounds, and characterization of the cases of equality in these bounds. A by-product is a sequential algorithm to calculate the exact maximum multiplicity by simple counting.

Estimation of the maximum multiplicity of an eigenvalue in terms of the vertex degrees of the graph of a matrix