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An important facet of the inverse eigenvalue problem for graphs is todetermine the minimum number of distinct eigenvalues of a particular graph. Weresolve this question for the join of a connected graph with a path. We thenfocus on bordering a matrix and attempt to control the change in the number ofdistinct eigenvalues induced by this operation. By applying borderingtechniques to the join of graphs, we obtain numerous results on the nature ofthe minimum number of distinct eigenvalues as vertices are joined to a fixedgraph.

Bordering of Symmetric Matrices and an Application to the Minimum Number of Distinct Eigenvalues for the Join of Graphs