The uncountable Hadwiger conjecture and characterizations of trees using graphs

We prove that the existence of a non-special tree of size \(\lambda\) $\lambda$ is equivalent to the existence of an uncountably chromatic graph with no \(K_{\omega1}\) $K_{\omega 1}$ minor of size \(\lambda\) $\lambda$ , 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.