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Title: Distinct Eignvalues and Sensitivity

Abstract: 

For a graph $G$, the class of real-valued symmetric matrices whose zero-nonzero pattern of off-diagonal entries is described by the adjacencies in $G$ is denoted by $S(G)$. The inverse eigenvalue problem for the multiplicities of the eigenvalues of $G$ is to determine for which ordered list of positive integers $m_1\geq m_2\geq \cdots\geq m_k$ with $\sum_{i=1}^{k} m_i=|V(G)|$, there exists a matrix in $S(G)$ with distinct eigenvalues ${\lambda_1,\lambda_2,\cdots, \lambda_k}$ such that $\lambda_i$ has multiplicity $m_i$.