Let \(G=(V,E)\) be a locally finite connected graph. We develop the first eigenvalue method on G introduced in 1963 by Kaplan [22] on Euclidean space, the discrete Phragmén-Lindelöf principle of parabolic equations and upper and lower solutions method on G. Using these methods, we establish the estimates and asymptotic behaviour of the life span of solutions to a semilinear heat equation with initial data \(\lambda \psi (x)\) for different scales of \(\lambda \) on G under some different conditions. Our results are different from the continuous case, which is related to the structure of the graph G.
