Edge-isoperimetric inequalities for the symmetric product of graphs

The k-th symmetric product of a graph G with vertex set V with edge set E is a graph G{k} with vertices as k-sets of V , where two k-sets are connected by an edge if and only if their symmetric difference is an edge in E . Using the isoperimetric properties of the vertex-induced subgraphs of G and Sobolev inequalities on graphs, we obtain various edge-isoperimetric inequalities pertaining to the symmetric product of certain families of finite and infinite graphs.