Sure, a graph is connected if it has a spanning tree, which requires |E|>=|V|-1. Presumably a graph is only highly connected if it has more edges than this mini

Sure, a graph is connected if it has a spanning tree, which requires |E|>=|V|-1. Presumably a graph is only highly connected if it has more edges than this minimal requirement, so we have |E|>=|V|, and this graph must have a loop.