The simplex graph
of an undirected graph is the graph with vertices given by the cliques
of and edges between pairs of cliques that
differ by insertion/deletion of exactly one vertex (Bandelt and van de Vel 1989,
Imrich et al. 1999, Alikhani and Ghanbari 2024).

The simplex graph of every graph is a bipartite graph
(Alikhani and Ghanbari 2024).

The following table sumamrizes the simplex graphs for some indexed families of graph.

Alikhani, S. and Ghanbari, N. "Golden Ratio in Graph Theory: A Survey." 9 Jul 2024. https://arxiv.org/abs/2407.15860.Bandelt,
H.-J. and van de Vel, M. "'Embedding Topological Median Algebras in Products
of Dendrons." Proc. London Math. Soc.58, 439-453, 1989.Imrich,
W.; Klavžar, S.; and Mulder, H. M. "Median Graphs and Triangle-Free
Graphs." SIAM J. Disc. Math.12, 111-118, 1999.