Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank operator A, it is shown that ‖A‖p≤(rankA)1/2p‖A‖2p≤(rankA)(2p−1)/2p2‖A‖2p2,for all p≥1 where ‖⋅‖p is the Schatten p-norm. If {λn(A)} is a listing of all non-zero eigenvalues (with multiplicity) of a compact operator A, then we show that [Formula presented] which improves the classical Weyl’s inequality ∑n|λn(A)|p≤‖A‖pp [Proc. Nat. Acad. Sci. USA 1949]. For an n×n matrix A, we show that the function p→n−1/p‖A‖p is monotone increasing on p≥1, complementing the well known decreasing nature of p→‖A‖p. As an application of these inequalities, we provide an upper bound for the sum of the absolute values of the zeros of a complex polynomial. As another application we provide a refined upper bound for the energy of a graph G, namely, E(G)≤2m(rank Adj(G)), where m is the number of edges, improving on a bound by McClelland in 1971.