Characterizing simplex graphs

The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form a subclass of median graphs and include many well-known families of graphs, such as gear graphs, Fibonacci cubes and Lucas cubes. In this paper, we characterize simplex graphs from four different perspectives: the first focuses on a graph class associated with downwards-closed sets -- namely, the daisy cubes; the second identifies all forbidden partial cube-minors of simplex graphs; the third is from the perspective of the $\Theta$ equivalent classes; and the fourth explores the relationship between the maximum degree and the isometric dimension. Furthermore, very recently, Betre et al.\ [K. H. Betre, Y. X. Zhang, C. Edmond, Pure simplicial and clique complexes with a fixed number of facets, 2024, arXiv: 2411.12945v1] proved that an abstract simplicial complex (i.e., an independence system) of a finite set can be represented to a clique complex of a graph if and only if it satisfies the Weak Median Property. As a corollary, we rederive this result by using the graph-theoretical method.