Recent years have witnessed a thriving growth of computing facilities connected at the network edge, cultivating edge networks as a fundamental infrastructure for supporting miscellaneous intelligent services.Meanwhile, Artificial Intelligence (AI) frontiers have extrapolated to the graph domain and promoted Graph Intelligence (GI). Given the inherent relation between graphs and networks, the interdiscipline of graph learning and edge networks, i.e., Edge GI or EGI, has revealed a novel interplay between them -- GI aids in optimizing edge networks, while edge networks facilitate GI model deployment. Driven by this delicate closed-loop, EGI is recognized as a promising solution to fully unleash the potential of edge computing power and is garnering growing attention. Nevertheless, research on EGI remains nascent, and there is a soaring demand within both the communications and AI communities for a dedicated venue to share recent advancements. To this end, this paper promotes the concept of EGI, explores its scope and core principles, and conducts a comprehensive survey concerning recent research efforts on this emerging field. Specifically, this paper introduces and discusses: 1) fundamentals of edge computing and graph learning,2) emerging techniques centering on the closed loop between graph intelligence and edge networks, and 3) open challenges and research opportunities of future EGI. By bridging the gap across communication, networking, and graph learning areas, we believe that this survey can garner increased attention, foster meaningful discussions, and inspire further research ideas in EGI.