Defining the connectivity of a graph

Question

Consider a simple binary undirected graph, which adjacency matrix is $A = \{a_{i,j}\} \in \{0, 1\}^{N \times N}.$

Suppose that all vertices of such graph have at least one neighbors, i.e.

$$k_i = \sum_{j=1}^N{a_{i,j}} \geq 1.$$

How do we call this graph?

Reading the definition of connectivity, I did not find anything about this particular case.

I would say that it is "minimally connected".

Is there a proper nomenclature for these graphs?

Answer 1

Requiring that every vertex have positive degree is not sufficient to make the graph connected. You may describe your graph as having no isolated vertices.