Gaussian upper heat kernel bounds and Faber-Krahn inequalities on graphs

We investigate the equivalence of relative Faber-Krahn inequalities and the conjunction of Gaussian upper heat kernel bounds and volume doubling on large scales on graphs. For the normalizing measure, we obtain their equivalence up to constants by imposing comparability of small balls and the vertex degree at their centers. Removing this comparability assumption on the cost of a local regularity condition entering the equivalence and allowing for a variable dimension lead to a further generalization. The variable dimension converges to the doubling dimension for increasing ball radius. If the counting measure or arbitrary measures are considered, the local regularity condition contains the vertex degree. Furthermore, correction functions for the Gaussian, doubling, and Faber-Krahn dimension depending on the vertex degree are introduced. For the Gaussian and doubling, the variable correction functions always tend to one at infinity. Moreover, the variable Faber-Krahn dimension can be related to the doubling dimension and the vertex degree growth.