Topological invariance under line graph transformations (Q350610)

Summary: It is shown that the line graph transformation \(G\mapsto L(G)\) of a graph \(G\) preserves an isomorphic copy of \(G\) as the nerve of a finite simplicial complex \(\mathbf K\) which is naturally associated with the Krausz decomposition of \(L(G)\). As a consequence, the homology of \(\mathbf K\) is isomorphic to that of \(G\). This homology invariance algebraically confirms several well known graph theoretic properties of line graphs and formally establishes the Euler characteristic of \(G\) as a line graph transformation invariant.