For any simple graph with finite set of verticies we assign a set of real symmetric matrices, whose $(i,j)$th entry is non-zero whenever $i \ne j$ and $\{i,j\}$ is an edge in $G$. We define maximum multiplicity of eigenvalues of a graph to be the largest possible multiplicity of eigenvalues of matrices in that set. We denote this parameter by $M(G)$. We also define parameter $Z(G)$ and show that for any simple graph $G$, $M(G)\leq Z(G)$. We take a closer look at graphs with cut-vertices and study parameters $M(G)$ and $Z(G)$ for these graphs.
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