On-diagonal lower estimate of heat kernels for locally finite graphs and its application to the semilinear heat equations

In this paper we establish a new on-diagonal lower estimate of heat kernels for connected, weighted, locally finite graphs. The result is then used to deal with the nonexistence of global solutions for a semilinear heat equation on locally finite graphs. Our results provide remarkable improvements to the work that was done recently by Lin and Wu (2017) [8,18].