U in your example are matrices, not graphs. You can use
AdjacencyGraph to create a graph from them.
The output of
GraphPlot is a
Graphics expression, not a
We use the term "graph expression" to denote the data structure that Mathematica uses to represent graphs.
A graph expression will always have
True if and only if
g is a graph expression. Usually, we simply say that "
g is a graph" (not a matrix, not a list of edges, not a picture showing a graph, but a graph).
Almost all functions whose name ends in
Q always return either
False in Mathematica. Thus,
True for a connected graph
False for a non-connected graph
False (without an error message!) for anything that is not a graph