The linear chains and monocyclic rings with carbon atoms fewer than 26 were studied at B3LYP/6-311$G^*$ level. The change trends that the stabilities of linear chains and monocyclic rings increase with the enlargement of cluster sizes were obtained, respectively. The theoretical calculations for carbon clusters lead to the conclusions that the small linear isomers of $C_3$~$C_9$ are more stable than the cyclic isomers and the monocyclic isomers contrarily become more stable than the linear isomers as cluster size is larger than $C_{10},$ which are consistent with those of early UPS and ion chromatography. Comparing with the stabilities of triplet and singlet states of linear clusters, it was confirmed that the triplet states of even-numbered clusters have higher stabilities, and so do the singlet states of odd-numbered clusters contradictorily. The average energies per atom of even-numbered monocyclic rings indicate that $(4m+2)$-numbered series are aromatic and $4m$-numbered series are anti-aromatic. The total energy comparisons give the conclusion that the 2D tetracyclic cluster and the cycloadduts with 24 carbon atoms are absolutely less stable than monocyclic isomer and three fullerenes. Several 3D polycyclic clusters and fullerene with $D_{6d}$ symmetry have middle stabilities. In addition, the theoretical IR spectra of linear chains and odd-numbered monocyclic rings were predicted.