This talk is devoted to semilinear parabolic equations on infinite weighted combinatorial graphs. We will address finite-time blow-up phenomena, considering both arbitrary initial data and sufficiently large initial data, and we will also discuss global existence for sufficiently small initial data. The results presented were obtained recently jointly with G. Grillo and G. Meglioli, and with F. Zucchero. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.