Is a circuit-free graph with n vertices and at least n − 1 edges connected? Why?

To check:

Whether a circuit free graph with n vertices and at least n−1 edges is connected or not.

Given information:

A circuit free graph with n vertices and at least n−1 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 n vertices and n−1 edges. Let G1,G2,.....,Gk be the connected component of G. To show that G is connected, we will show that k=1.

Each graph Gi is a tree since each Gi is connected and circuit free.

For each i=1,2,......,k let Gi have ni vertices. Then,

n1+n2+....+nk=n

For a positive integer n any tree with n vertices has n−1 edges.

Thus, G1 has n1−1 edges.

G2 has n2−1 edges