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Extended Wenger Graphs

Abstract

Wenger graphs were originally introduced as examples of dense graphs that do not have cycles of a given size. Graphs with similar properties were known at the time, but Wenger graphs are based on algebraic relations in finite fields, and as such are easier to understand and analyze.

Wenger graphs are bipartite, with the vertices consisting of two copies of the vector space of dimension m+1 over the finite field of order q. These two sets of vertices are called points and lines, with a point vertex connected to a line vertex if the equations $p_k + l_k = l_1 f_k(p_1)$ are satisfied for k = 2, 3, ..., m+1. In the original Wenger graph, the function $f_k(x)$ was given by $f_k(x)=x^{k-1}$.

Since their introduction in 1991, the original Wenger graph concept has been extended to include linearized and jumped Wenger graphs, and some results are known for extensions in general. In this dissertation, another extension, the extended Wenger graph, is introduced and analyzed, and a new result about polynomial root patterns is proven.

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