  We prove that the existence of a non-special tree of size $\lambda$ is
equivalent to the existence of an uncountably chromatic graph with no
$K_{\omega_1}$ minor of size $\lambda$, establishing a connection between the
special tree number and the uncountable Hadwiger conjecture. Also
characterizations of Aronszajn, Kurepa and Suslin trees using graphs are
deduced. A new generalized notion of connectedness for graphs is introduced
using which we are able to characterize weakly compact cardinals.
