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We survey index theorems counting eigenvalues of linearized Hamiltonian systems and characteristic values of polynomial operator pencils. We present a simple common graphical interpretationand generalization of the index theory using the concept of graphical Krein signature. Furthermore, we prove that derivatives of an eigenvector u = u(λ) of an operator pencil L(λ) satisfying L(λ)u(λ) = µ(λ)u(λ) evaluated at a characteristic value of L(λ) do not only generate an arbitrary chain of root vectors of L(λ) but the chain that carries an extra information.

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