Homology of the line graph and of related graph-valued functions
Let \(G^{\uparrow}\) denote the simplicial complex having the vertex sets of all complete subgraphs of the finite graph G as simplices. Then the first Betti numbers of \(G^{\uparrow}\) and of the complex \(L(G)^{\uparrow}\) of its line graph coincide. The second Betti number of \(G^{\uparrow}\) is not greater than that of \(L(G)^{\uparrow}\). Furthermore, the n-th Betti number of \(L(G)^{\uparrow}\) vanishes for every \(n\geq 3\). Similar or related results are obtained for the total graph, the middle graph, the n-simplex graph, the clique-line graph, and the line-clique graph.
scientific article; zbMATH DE number 4139794