  We derive Gaussian heat kernel bounds on graphs with respect to a fixed
origin for large times under the assumption of a Sobolev inequality and volume
doubling on large balls. The upper bound from our previous work [KR22] is
affected by a new correction term measuring the distance to the origin. The
main result is then applied to anti-trees with unbounded vertex degree,
yielding Gaussian upper bounds for this class of graphs for the first time. In
order to prove this, we show that isoperimetric estimates with respect to
intrinsic metrics yield Sobolev inequalities. Finally, we prove that anti-trees
are Ahlfors regular and that they satisfy an isoperimetric inequality of a
larger dimension.
