The Alternating and Adjacency Polynomials, and Their Relation with the Spectra and Diameters of Graphs

Let Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ1 > λ2 > · · · > λd. For every k = 0, 1, . . . , d−1, the k-alternating polynomial Pk is defined to be the polynomial of degree k and norm ‖Pk‖∞ = max1≤l≤d{|Pk(λl)|} = 1 that attains maximum value at λ. These polynomials, which may be thought of as the discrete version of the Chebychev ones, were recently used by the authors to bound the diameter D(Γ) of Γ in terms of its eigenvalues. Namely, it was shown that Pk(λ) > ‖ν‖ − 1 ⇒ D(Γ) ≤ k, where ν is the (positive) eigenvector associated to λ with minimum component 1. In this work we improve upon this result by assuming that some extra information about the structure of Γ is known. To this end, we introduce the so-called τ -adjacency polynomial Qτ . For each 0 ≤ τ ≤ d, the polynomial Qτ is defined to be the polynomial of degree τ and norm ‖Qτ‖A = max1≤i≤n{‖Qτ (A)ei‖} = 1 that attains maximum value at λ. Then it is shown that Pk(λ) > ‖ν‖/Qτ (λ)−1⇒ D(Γ) ≤ k+2τ . Some applications of the above results, together with new bounds for generalized diameters, are also presented.