Transfermium nuclei (101$leq$Z$leq$110) are investigated thoroughly to describe structural properties viz. deformation, radii, shapes, magicity, etc. as well as their probable decay chains. These properties are explored using relativistic mean-field (RMF) approach and compared with other theories along with available experimental data. Neutron numbers N$=$152 and 162 have come forth with a deformed shell gap whereas N$=$184 is ensured as a spherical magic number. The region with N$>$168 bears witness of the phenomenon of shape transition and shape coexistence for all the considered isotopic chains. Experimental $alpha$-decay half-lives are compared with our theoretical half-lives obtained by using various empirical/semi-empirical formulas. The recent formula proposed by Manjunatha textit{et al.}, which results best among the considered 10 formulas, is further modified by adding asymmetry dependent terms ($I$ and $I^2$). This modified Manjunatha formula is utilized to predict probable $alpha$-decay chains that are found in excellent agreement with available experimental data.
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