  We define a broad class of graphs that generalize the Gordian graph of knots.
These knot graphs take into account unknotting operations, the concordance
relation, and equivalence relations generated by knot invariants. We prove that
overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception
of a particular family of quotient knot graphs. We also investigate the
property of homogeneity, and prove that the concordance knot graph is
homogeneous. Finally, we prove that that for any $n$, there exists a knot $K$
such that the ball of radius $n$ in the Gordian graph centered at $K$ contains
no connected sum of torus knots.
