On monomial graphs of girth eight

  • Authors:

  • Vasyl Dmytrenko;Felix Lazebnik;Jason Williford

  • Affiliations:

  • Department of Mathematics, Temple University, Philadelphia, PA 19122, USA;Department of Mathematical Science, Ewing Hall, University of Delaware, Newark, DE 19716, USA;Department of Mathematical Sciences, Worcester Polytechnic Institute, 100, Institute road, Worcester, MA 01609-2280, USA

  • Venue:
  • Finite Fields and Their Applications
  • Year:
  • 2007

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Abstract

Let e be a positive integer, p be an odd prime, q=p^e, and F"q be the finite field of q elements. Let f"2,f"3@?F"q[x,y]. The graph G=G"q(f"2,f"3) is a bipartite graph with vertex partitions P=F"q^3 and L=F"q^3, and edges defined as follows: a vertex (p)=(p"1,p"2,p"3)@?P is adjacent to a vertex [l]=[l"1,l"2,l"3] if and only ifp"2+l"2=f"2(p"1,l"1)andp"3+l"3=f"3(p"1,l"1). Motivated by some questions in finite geometry and extremal graph theory, we ask when G has no cycle of length less than eight, i.e., has girth at least eight. When f"2 and f"3 are monomials, we call G a monomial graph. We show that for p=5, and e=2^a3^b, a monomial graph of girth at least eight has to be isomorphic to the graph G"q(xy,xy^2), which is an induced subgraph of the classical generalized quadrangle W(q). For all other e, we show that a monomial graph is isomorphic to a graph G"q(xy,x^ky^2^k), with 1=