In a diamond, each carbon atom forms four covalent bonds with neighboring carbon atoms, resulting in a tetrahedral structure.

To elaborate, diamond is one of the allotropes of carbon, which are distinct structural forms in which carbon can exist. In the diamond structure, each carbon atom is covalently bonded to four other carbon atoms. These covalent bonds are exceptionally strong and arranged in a three-dimensional tetrahedral formation. This geometric arrangement leads to a rigid, lattice-like structure that contributes to the remarkable hardness of diamonds.

All carbon atoms in a diamond exhibit $sp^3$ hybridization. This process involves the combination of one $2s$ orbital and three $2p$ orbitals from each carbon atom to create four equivalent hybrid orbitals. These $sp^3$ hybrid orbitals are utilized to form the four covalent bonds with adjacent carbon atoms, ensuring that the bonds are symmetrically distributed around the carbon atom, thus establishing the tetrahedral geometry.

The exceptional strength of these covalent bonds, along with the rigidity of the resultant structure, accounts for the extraordinary hardness and durability of diamonds. Furthermore, this structural integrity contributes to their high melting point and excellent thermal conductivity. However, diamonds do not conduct electricity; this is due to the fact that all electrons are involved in covalent bonding, leaving no free electrons available to carry an electric charge.

In summary, the carbon atoms in a diamond are interconnected through robust covalent bonds in a tetrahedral arrangement. This unique structure, combined with the $sp^3$ hybridization of the carbon atoms, endows diamonds with their distinctive and valuable properties.