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 K-1+Pn, where K-1 = K-1<overline> is the singleton graph and P-n is the path on n >= 1 vertices, we show that QEC(K-1+P-n)=-alpha(n-2), where alpha(n) is the minimal zero of a new polynomial Phi(n)(x) related to Chebyshev polynomials of the second kind. Moreover, for an even n we have alpha n=min ev(An), 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)) <= alpha(n) <min ev(An). (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).