The discrete Chebyshev polynomials tn(x, N) are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the points x = 0, 1, … , N − 1, N being a fixed positive integer. By using a double integral representation, we derive two asymptotic expansions for tn(aN, N + 1) in the double scaling limit, namely, N →∞ and n/N → b, where b ∈ (0, 1) and a ∈ (−∞, ∞). One expansion involves the confluent hypergeometric function and holds uniformly for , and the other involves the Gamma function and holds uniformly for a ∈ (−∞, 0). Both intervals of validity of these two expansions can be extended slightly to include a neighborhood of the origin. Asymptotic expansions for  can be obtained via a symmetry relation of tn(aN, N + 1) with respect to . Asymptotic formulas for small and large zeros of tn(x, N + 1) are also given.

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Symmetric graphs of valencies 3, 4 and 5 and square-free order have been classified in the literature. In this paper, we will present a complete classification of symmetric graphs of square-free order and any prime valency which admit a soluble arc-transitive group, and a complete classification of 7-valent symmetric graphs of square-free order.

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