Boundary graphs: The limit case of a spectral property (Q1841901)

Let \(\mathbb{R}_k[x]\) be the set of real polynomials of degree \(k\). Let \(\lambda_0> \lambda_1>\cdots> \lambda_d\) be the distinct eigenvalues of a graph \(G\) of order \(n\). In [J. Comb. Theory, Ser. B 67, No. 1, 48-61 (1996; Zbl 0857.05101)] the authors introduced the alternating polynomials \(P_k\) such that \(P_k(\lambda_0)= \max_{p(x)\in \mathbb{R}_k[x]}\{p(\lambda_0)\mid \|p\|_\infty\leq 1\}\), where \(\|p\|_\infty= \max_{1\leq i\leq d}\{|p(\lambda_i)|\}\). In the present paper they discuss properties of alternating polynomials for boundary graphs.