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Let $X$ be a continuous time random walk on a weighted graph. Given theon-diagonal upper bounds of transition probabilities at two vertices $x_1$ and$x_2$, we use an adapted metric initiated by Davies, and obtain Gaussian upperestimates for the off-diagonal transition probability $P_{x_1}(X_t=x_2)$.

Pointwise upper estimates for transition probability of continuous time random walks on graphs