Graphs can effectively model biomolecules, computer systems, and other applications. A weighted graph is a graph in which values or labels are assigned to the edges of the graph. However, in this thesis, we assign values to the vertices of the graph rather than the edges and we modify several standard graphical measures to incorporate these vertex weights. In particular, we designate the degree of each vertex as its weight. Previous research has not investigated weighting vertices by degree. We find the vertex weighted domination number in connected graphs, beginning with trees, and we define how weighted vertices can affect eccentricity, independence number, and connectivity.