Graphite and diamond are carbon allotropes with different structures. Graphite has layered hexagonal rings with delocalized π-electrons, while diamond has a 3D tetrahedral network with localized σ-bonds.

Graphite has higher electrical conductivity due to mobile electrons: ${\sigma }_{\text{elec}}>{\sigma }_{\text{dia}}$. Its C-C bond order is 1.33 (from resonance), higher than diamond's 1. Diamond is harder due to strong covalent bonds in all directions: $H_{\text{dia}}>H_{\text{graph}}$. Diamond has higher thermal conductivity from efficient phonon transport.

Final answer: Only "Graphite has higher electrical conductivity" and "Graphite has higher C-C bond order" are correct.