# Spectral and Combinatorial Properties of Some Algebraically Defined Graphs

Let k> 3 be an integer, q be a prime power, and F_q denote the field of q elements. Let f_i, g_i∈F_q[X], 3< i< k, such that g_i(-X) = - g_i(X). We define a graph S(k,q) = S(k,q;f_3,g_3,...,f_k,g_k) as a graph with the vertex set F_q^k and edges defined as follows: vertices a = (a_1,a_2,...,a_k) and b = (b_1,b_2,...,b_k) are adjacent if a_1 b_1 and the following k-2 relations on their components hold: b_i-a_i = g_i(b_1-a_1)f_i(b_2-a_2/b_1-a_1) , 3< i< k. We show that graphs S(k,q) generalize several recently studied examples of regular expanders and can provide many new such examples.

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