Algebraic Properties of Clique Complexes of Line Graphs

Abstract

Let H be a simple undirected graph and G = L(H) be its line graph. Assume that ∆(G) denotes the clique complex of G. We show that ∆(G) is sequentially Cohen-Macaulay if and only if it is shellable if and only if it is vertex decomposable. Moreover if ∆(G) is pure, we prove that these conditions are also equivalent to being strongly connected. Furthermore, we state a complete characterizations of those H for which ∆(G) is Cohen-Macaulay, sequentially Cohen-Macaulay or Gorenstein. We use these characterizations to present linear time algorithms which take a graph G, check whether G is a line graph and if yes, decide if ∆(G) is Cohen-Macaulay or sequentially Cohen-Macaulay or Gorenstein.

Topology of Clique Complexes of Line Graphs

“…Clique complexes of graphs have a very rich literature in topological combinatorics, for instance see [1,10]. Recently, Nikseresht [17] studied some algebraic properties like Cohen-Macaulay, sequentially Cohen-Macaulay, Gorenstein, etc., of clique complexes of line graphs. In this article, we determine the exact homotopy type of ∆L(G) for various classes of graphs G. Moreover, we show that ∆L in each of these cases is homotopy equivalent to a wedge of equidimensional spheres.…”