When I learned graph theory, the prevailing understanding was that graphs are mathematical abstractions of networks. Most people have an intuitive understanding

When I learned graph theory, the prevailing understanding was that graphs are mathematical abstractions of networks. Most people have an intuitive understanding that there is some underlying principle of connectivity which make "computer networks" and "social networks" similar to each other in some way. So, focus on networks. If you're a graph theorist and need to talk to lay people, say that you research the underlying mathematics which combine computer networks and social networks into a single topic. You can then follow up by saying, "mathematicians also call this a graph, which has a specific formal definition, but for this conversation, let's just keep calling it a network, and we'll imagine every network can be represented as a bunch of dots connected by lines." Most lay people are more interested in applications than formal definitions and theorems, so give them some interesting but intuitive problems which are investigated in network theory, like travelling salesman or "chinese postman."