TITLE: What are some measures of connectedness in graphs?
QUESTION [18 upvotes]: I am not a mathematician (I am an engineer who is working on improving his mathematics), so I apologize in advance if my question is trivial.
Consider a graph of $N$ nodes, with some defined criterion as to whether two nodes are connected or not. What are some measures of the graph's connectedness? I can think of several such measures, but I'm not sure which ones make the most mathematical sense:

Whether the entire graph is connected or not. This is, however, a binary measure, and does not capture much information.
The size of the largest intra-connected sub graph ($0 \leq n \leq N$). This, however, tells you nothing about what's happening in the rest of the graph.
The number of intra-connected (but not inter-connected) sub-graphs in the graph ($0 \leq n \leq N$).
Each node $i\in \{1,2,\dots,N\}$ is assigned a 'score' $s_i\in\{1,2,\dots,N\}$, which measures how many other nodes it is connected to. The graphs' connectedness is then measured as the average of these scores, i.e. $\sum_i s_i / N$, such that it lies in the interval $[0,N]$.

The kind of questions I am interested in are ones like: Is 4. a mathematically sound measure? Can it give rise to anomalies? Are there better measures that are more robust?
Please forgive me if I am not rigorous enough in my explanations. As an engineer, I have been trained to think in 'intuitive' (rather than formal) terms, which can often be a huge help, but at other times a hindrance.

REPLY [14 votes]: The most common measures of connectivity are edge-connectivity and vertex-connectivity. The vertex-connectivity, or just connectivity, of a graph is the minimum number of vertices you have to remove before you can even hope to disconnect the graph. A graph is called $k$-vertex-connected, or just $k$-connected, if its connectivity is at least $k$. Edge-connectivity and $k$-edge-connected are defined similarly.
As an example, suppose we've got a tree $T$ with at least 3 vertices. In a tree, any two vertices are connected by exactly one path. In particular, removing any internal (i.e. not one of the endpoints) part of any path from $T$ disconnects the graph. Thus, $T$ has vertex- and edge-connectivity $1$. (We needed at least 3 vertices here to guarantee that we can find a path with interior vertices, though with 2 vertices we still have edge-connectivity $1$. By convention, the tree with two vertices has connectivity $1$.)
As another example, one can show (try it!) that a cycle of length greater than 3 has edge- and vertex-connectivity $2$. (As with trees, for a 3-cycle---there's no such thing as a 1- or 2-cycle---, the edge-connectivity still makes sense and is $2$, but for vertex-connectivity we have to resort to convention: we simply assign it connectivity $2$).

REPLY [10 votes]: Sometimes in math turning the question on its head leads to useful definitions.  In this case we might ask how few edges are necessary to remove in order to disconnect a graph.  If no single edge's removal gives a disconnected graph, we say the graph is doubly connected, etc.
There's a topic called community detection which in loose terms tries to identify vertex subsets of a graph that are more closely connected within themselves than to the other portions.