  In this work we consider the Keller-Segel model for chemotaxis on networks,
both in the doubly parabolic case and in the parabolic-elliptic one.
Introducing appropriate transition conditions at vertices, we prove the
existence of a time global and spatially continuous solution for each of the
two systems. The main tool is the use of the explicit formula for the
fundamental solution of the heat equation on a weighted graph and of the
corresponding sharp estimates.
