A circuit-free graph has ten vertices and nine edges. Is it connected? Why?

To check:

Whether a circuit free graph, having ten vertices and nine edges, is connected or not.

Given information:

Graph with 10 vertices  and 9 edges.

Concept used:

Any tree with n vertices has n−1 edges and total degree is 2(n−1).

Calculation:

Suppose G is a circuit free graph with ten vertices and nine edges.

Let G1,G2,.....,Gk be the connected components of G.

To show that G is connected, we will show that k=1.

Each Gi is a tree since each Gi has ni vertices.

Note that since G has ten vertices in all,

n1+n2+......+nk=10

Since for any positive integer n, any tree with n vertices has n−1 edges, we have G1 and n1−1 edges, G2 has n2−1 edges,