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The goal of this article is to display a Q -polynomial structure for the Attenuated Space poset A q (N , M). The poset A q (N , M) is briefly described as follows. Start with an (N + M) -dimensional vector space H over a finite field with q elements. Fix an M -dimensional subspace h of H. The vertex set X of A q (N , M) consists of the subspaces of H that have zero intersection with h. The partial order on X is the inclusion relation. The Q -polynomial structure involves two matrices A , A ⁎ ∈ Mat X (C) with the following entries. For y , z ∈ X the matrix A has (y , z) -entry 1 (if y covers z); q dim y (if z covers y); and 0 (if neither of y , z covers the other). The matrix A ⁎ is diagonal, with (y , y) -entry q − dim y for all y ∈ X. By construction, A ⁎ has N + 1 eigenspaces. By construction, A acts on these eigenspaces in a (block) tridiagonal fashion. We show that A is diagonalizable, with 2 N + 1 eigenspaces. We show that A ⁎ acts on these eigenspaces in a (block) tridiagonal fashion. Using this action, we show that A is Q -polynomial. We show that A , A ⁎ satisfy a pair of relations called the tridiagonal relations. We consider the subalgebra T of Mat X (C) generated by A , A ⁎. We show that A , A ⁎ act on each irreducible T -module as a Leonard pair. [ABSTRACT FROM AUTHOR]

A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph K n ∖ C k , obtained from the complete graph K n with n vertices by deleting all edges of a cycle C k with k vertices, is determined by its spectrum for k ∈ { 3 , 4 , 5 } , but not for k = 6. In this paper, we show that k = 6 is the unique exception for the spectral determination of K n ∖ C k. [ABSTRACT FROM AUTHOR]