Quadratic embedding constants of fan graphs and graph joins

We derive a general formula for the quadratic embedding constant of a graph join ${\bar{K}}_m+G$, where ${\bar{K}}_m$ is the empty graph on $m\ge 1$ vertices and $G$ is an arbitrary graph. Applying our formula to a fan graph $K_1+P_n$, where $K_1={\bar{K}}_1$ is the singleton graph and $P_n$ is the path on $n\ge 1$ vertices, we show that $\mathrm{Q}\mathrm{E}\mathrm{C}(K_1+P_n)=-{\tilde{\alpha }}_n-2$, where ${\tilde{\alpha }}_n$ is the minimal zero of a new polynomial ${\mathrm{\Phi }}_n(x)$ related to Chebyshev polynomials of the second kind. Moreover, for an even $n$ we have ${\tilde{\alpha }}_n=\min \mathrm{e}\mathrm{v}(A_n)$, where the right-hand side is the An minimal eigenvalue of the adjacency matrix $A_n$ of $P_n$. For an odd $n$ we show that $\min \mathrm{e}\mathrm{v}(A_{n+1})\le {\tilde{\alpha }}_n<\min \mathrm{e}\mathrm{v}(A_n)$.