The correct IUPAC name of the following compound is :

1. Identify the longest continuous carbon chain: The longest chain in the given structure contains $7$ carbon atoms. Therefore, the parent alkane is heptane.

2. Identify the substituents: There is an ethyl group ($\text{-CH}_2\text{CH}_3$) and a methyl group ($\text{-CH}_3$) attached to the main chain.

3. Number the carbon chain:
- Numbering from left to right gives the substituents at positions $3$ (ethyl)$and 5$ (methyl).
- Numbering from right to left gives the substituents at positions $3$ (methyl)$and 5$ (ethyl).

4. Apply the alphabetical rule: When two different numbering schemes give the same set of locants ($3, 5$), the lower number is assigned to the substituent that is cited first alphabetically. Since 'ethyl' comes before 'methyl', the ethyl group must get the lower number ($3$).

5. Formulate the name: Combining these parts, the correct IUPAC name is $3$-ethyl-$5$-methylheptane.

Answer: $3$-ethyl-$5$-methylheptane