The minimum number of elements that must be added to the relation $R=\left\{\right(1,2),(2,3\left)\right\}$ on the set $\{1,2,3\}$ So that it is an equivalence relation.

The given relation $R=\left\{\left(1,2\right),\left(2,3\right)\right\}$

To make this relation is Reflexive relation add the ordered pairs $\left(1,1\right),\left(2,2\right),\left(3,3\right)$

Now the relation is $R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,3\right)\right\}$

To make the above relation symmetric add the ordered pairs $\left(2,1\right),\left(3,2\right)$

Now the relation is $R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,3\right),\left(2,1\right),\left(3,2\right)\right\}$

To make the above relation trasitive, add the ordered pairs $\left(1,3\right),\left(3,1\right)$,

New relation  is $R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,3\right),\left(2,1\right),\left(3,2\right),\left(1,3\right),\left(3,1\right)\right\}$

Therefore, the total number of ordered pairs added to $R=\left\{\left(1,2\right),\left(2,3\right)\right\}$ to make the relation $R$ is equivalence is $\boxed{\mathbf{7}}$