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This paper addresses the following three topics: positive semidefinite (psd)matrix completions, universal rigidity of frameworks, and the Strong ArnoldProperty (SAP). We show some strong connections among these topics, usingsemidefinite programming as unifying theme. Our main contribution is asufficient condition for constructing partial psd matrices which admit a uniquecompletion to a full psd matrix. Such partial matrices are an essential tool inthe study of the Gram dimension $\gd(G)$ of a graph $G$, a recently studiedgraph parameter related to the low psd matrix completion problem. Additionally,we derive an elementary proof of Connelly's sufficient condition for universalrigidity of tensegrity frameworks and we investigate the links between thesetwo sufficient conditions. We also give a geometric characterization of psdmatrices satisfying the Strong Arnold Property in terms of nondegeneracy of anassociated semidefinite program, which we use to establish some links betweenthe Gram dimension $\gd(\cdot)$ and the Colin de Verdi\`ere type graphparameter $\nu^=(\cdot)$.

Positive Semidefinite Matrix Completion, Universal Rigidity and the Strong Arnold Property