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We simulated heat propagation in the integrated graphene heat spreaders within the interconnect hierarchy. In the considered design, the graphene layers perform the dual functions of interconnects and heat spreaders. We investigated Joule heating effects within the chip with graphene interconnect networks and heat spreaders. Numerical solutions for direct current and heat propagation equations were found using the finite-element method. The simulation results showed that the use of graphene as interconnects as well as heat spreaders lowers the maximum temperature of the chip. The maximum temperature of the chip was studied as a function of the interconnect current and thickness of few-layer graphene. Our results are important for design of graphene-based thermal and electrical interconnect networks in the next generations of integrated circuits and 3-D electronics.