On a class of polynomials and its relation with the spectra and diameters of graphs
Let \(\lambda_1,\lambda_2,\dots,\lambda_d\) be reals with \(\lambda_1>\lambda_2>\cdots>\lambda_d\). For every \(k=1,2,\dots,d\), the \(k\)-alternating polynomial \(P_k\) is the polynomial of degree \(k\) and norm \(|P_k|_\infty= \max_{1\leq i\leq d}\{|P_k(\lambda_i)|\}\leq 1\) that attains maximum absolute value at any point \(\lambda\not\in[\lambda_d,\lambda_1]\). These polynomials can be considered as the discrete version of Chebychev polynomials. Some basic properties of \(P_k\) are studied, and it is shown how to compute them in general. The results are applied to the study of the relationship between the (standard or Laplacian) spectrum of a graph or bipartite graph and its diameter, improving previous results.
scientific article; zbMATH DE number 934804