Connection Laplacian, Hodge Laplacian, and Tensor Laplacian of a Graph

The graph Laplacian is a linear operator that takes real-valued functions on vertices to real-valued functions on vertices, or alternatively, a bilinear functional on pairs of real-valued functions on vertices. Three generalizations have recently been studied: the connection Laplacian [Wu-Singer, 2012] allows for tensor-valued or Lie group-valued functions on vertices; the Hodge Laplacian [Jiang-Lim-Yao-Ye, 2011] allows for real-valued functions on higher-order cliques (vertices are 0-cliques); the tensor Laplacian [Hu-Qi, 2013] allows for k-linear functional on k-tuples of real-valued vertex functions. We will compare the spectral theories of these three Laplacians in the context of graph theory.