Determine all graphs with matching number = 1 Determine all graphs G, without isolated vertices, such that the
matching number=1.

Could anyone help me this question?
I am really confused about it. 

How to determine all possible graphs with this condition?
If your graph is to have matching number 1, you need at least one edge $e_1$, between vertices, say, $v_1$ and $v_2$. This certainly works.

Can we add more things? If we add a new vertex $v_3$, it is required that $v_3$ is not isolated, so, it is connected to something. If it is a new vertex, then our matching number is at least 2. So we can add an extra $v_3$ and assume WLOG that it is connected to $v_1$. (This gives us the path on 3 vertices).

Edit: Here I missed that you can join $v_3$ to BOTH $v_1$ and $v_2$, so, we have $P_3$ (the path on 3 vertices) and $K_3$ (the complete graph on 3 vertices). We can't add any new edges to $K_3$, so the next paragraph is assuming we are dealing with $P_3$.

Can we add more? If we want to add $v_4$, for the same reason as above, it can't be isolated, and can't be connected to something new, so it has to be connected to one of our $v_1,v_2,v_3$. If it is connected to $v_2$ or $v_3$, then our graph contains a path on 4 vertices, which has matching number 2. So, any new edge must be connected to $v_1$.

Continuing in this fashion, you obtain that larger graphs must have a central vertex $v_1$, so that every other vertex is connected to it, and there are no other edges. I'm not sure if such graphs have a name, but I would guess "star-shaped".

You should be able to turn the above intuition into a proof.

Edit: I am told that these are called Star-graphs http://en.wikipedia.org/wiki/Star_(graph_theory)
There is only one family of bipartite graphs for which the matching number is 1. Its commonly denoted as K_{1,r}, r \geq 1. In this graph, one partite set has exactly one vertex, say 'v'; and the other partite set has 'r' vertices, all of which are connected to the one vertex 'v'.