Quadratic embedding constants of fan graphs and graph joins

Abstract

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