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) is an element of 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) not equal 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 the graphs S(k, q) generalize several recently studied examples of regular expanders and can provide many new such examples.