Csikvari Peter:
Graph polynomials and graph generating functions
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In the talk we will study some analytic properties of the so-called
independence polynomial and the adjoint polynomial. Concerning the
independence polynomial we will give a new proof for the fact that it
has a unique root of smallest modulus which is real. In case of the
adjoint polynomial we will prove that its root having the largest
absolute value is real and it is at most 4(D-1), where $D$ is the
largest degree. This bound is sharp. In the first half of the talk I
will try to explain the connection of these results with extremal graph
theory and statistical physics.