On the uniqueness of some girth eight algebraically defined graphs

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• 作者：Brian G. Kronenthal^a ; ^{kronenthal@kutztown.edu" class="auth_mail" title="E-mail the corresponding author} ; Felix Lazebnik^b ; ^{lazebnik@math.udel.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Algebraically defined graph ; Cycle ; Girth eight ; Lefschetz principle ; Finite field ; Generalized quadrangle
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：19 June 2016
• 年：2016
• 卷：206
• 期：Complete
• 页码：188-194
• 全文大小：387 K

文摘

Let F be a field. For a polynomial f∈F[x,y], we define a bipartite graph Γ_F(f) with vertex partition P∪L$P\cup L$, P=F³=L, and (p₁,p₂,p₃)∈P$(p_1,p_2,p_3)\in P$ is adjacent to [l₁,l₂,l₃]∈L$[l_1,l_2,l_3]\in L$ if and only if It is known that the graph Γ_F(xy²) has no cycles of length less than eight. The main result of this paper is that Γ_F(xy²) is the only graph Γ_F(f) with this property when F is an algebraically closed field of characteristic zero; i.e. over such a field F, every graph Γ_F(f) with no cycles of length less than eight is isomorphic to Γ_F(xy²). We also prove related uniqueness results for some polynomials f$f$ over infinite families of finite fields.