Proof of a conjecture on monomial graphs |
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Affiliation: | 1. Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, United States;2. Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, United States |
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Abstract: | Let e be a positive integer, p be an odd prime, , and be the finite field of q elements. Let . The graph is a bipartite graph with vertex partitions and , and edges defined as follows: a vertex is adjacent to a vertex if and only if and . If and , the graph contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs containing no cycles of length less than eight and not isomorphic to the graph , even without requiring them to be edge-transitive. So far, no such graphs have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture. |
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Keywords: | Generalized quadrangle Girth eight Monomial graph Permutation polynomial Power sum |
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