The maximum multiplicity of an eigenvalue of symmetric matrices with a   given graph
  For a graph G, M(G) denotes the maximum multiplicity occurring of an eigenvalue of a symmetric matrix whose zero-nonzero pattern is given by edges of G. We introduce two combinatorial graph parameters T^-(G) and T^+(G) that give a lower and an upper bound for M(G) respectively, and we show that these bounds are sharp. 
Keivan Hassani Monfared and Sudipta Mallik
math.CO