Quadratic embedding constants of fan graphs and graph joins

We derive a general formula for the quadratic embedding constant of a graph join K¯_m+G, where K¯_m is the empty graph on m≥1 vertices and G is an arbitrary graph. Applying our formula to a fan graph K₁+P_n, where K₁=K¯₁ is the singleton graph and P_n is the path on n≥1 vertices, we show that QEC(K₁+P_n)=−α˜_n−2, where α˜_n is the minimal zero of a new polynomial Φ_n(x) related to Chebyshev polynomials of the second kind. Moreover, for an even n we have α˜_n=min⁡ev(A_n), 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⁡ev(A_n+1)≤α˜_n<min⁡ev(A_n).