A relation between the cube polynomials of partial cubes and the clique polynomials of their crossing graphs

Partial cubes are the graphs which can be embedded into hypercubes. The cube polynomial of a graph $G$ is a counting polynomial of induced hypercubes of $G$, which is defined as $C\left(G,x\right)≔{\sum }_{i⩾0}{\alpha }_i\left(G\right)x^i$, where ${\alpha }_i\left(G\right)$ is the number of induced $i$-cubes (hypercubes of dimension $i$) of $G$. The clique polynomial of $G$ is defined as $Cl\left(G,x\right)≔{\sum }_{i⩾0}{a}_i\left(G\right)x^i$, where $a_i\left(G\right)$ ($i⩾1$) is the number of $i$-cliques in $G$ and $a_0\left(G\right)=1$. Equivalently, $Cl\left(G,x\right)$ is exactly the independence polynomial of the complement $\overline{G}$ of $G$. The crossing graph $G^{\#}$ of a partial cube $G$ is the graph whose vertices are corresponding to the $\Theta$-classes of $G$, and two $\Theta$-classes are adjacent in $G^{\#}$ if and only if they cross in $G$. In the present paper, we prove that for a partial cube $G$, $C\left(G,x\right)⩽Cl\left(G^{\#},x+1\right)$ and the equality holds if and only if $G$ is a median graph. Since every graph can be represented as the crossing graph of a median graph, the above necessary-and-sufficient result shows that the study on the cube polynomials of median graphs can be transformed to the one on the clique polynomials of general graphs (equivalently, on the independence polynomials of their complements). In addition, we disprove the conjecture that the cube polynomials of median graphs are unimodal.