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We introduce a new one-variable polynomial invariant of graphs, which we call
the skew characteristic polynomial. For an oriented simple graph, this is just
the characteristic polynomial of its anti-symmetric adjacency matrix. For
nonoriented simple graphs the definition is different, but for a certain class
of graphs (namely, for intersection graphs of chord diagrams), it gives the
same answer if we endow such a graph with an orientation induced by the chord
diagram. We prove that this invariant satisfies Vassiliev's $4$-term relations
and determines therefore a finite type knot invariant. We investigate the
behaviour of the polynomial with respect to the Hopf algebra structure on the
space of graphs and show that it takes a constant value on any primitive
element in this Hopf algebra. We also provide a two-variable extension of the
skew characteristic polynomial to embedded graphs and delta-matroids. The
$4$-term relations for the extended polynomial prove that it determines a
finite type invariant of multicomponent links.