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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si1.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=f72988a8747a3b8bffaf5f249699d114" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">$\mathbb{F}$ be a field. For a polynomial class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si2.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=5cce17c4aa007525aeae5b25b46ff7ce" title="Click to view the MathML source">f∈F[x,y]class="mathContainer hidden">class="mathCode">$f\in \mathbb{F}[x,y]$, we define a bipartite graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si3.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=2722c2becd8df07ccb26ceb9892ccba9" title="Click to view the MathML source">Γ_F(f)class="mathContainer hidden">class="mathCode">${\Gamma }_{\mathbb{F}}\left(f\right)$ with vertex partition class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si4.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=0b95bc2832a8938dde3d2bf945fa1b41" title="Click to view the MathML source">P∪Lclass="mathContainer hidden">class="mathCode">$P\cup L$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si5.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=6ee7dbc66a7b33ff297d182cd545cf08" title="Click to view the MathML source">P=F³=Lclass="mathContainer hidden">class="mathCode">$P={\mathbb{F}}^3=L$, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si6.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=63de7eb4b09215f129c1ce908bdee93d" title="Click to view the MathML source">(p₁,p₂,p₃)∈Pclass="mathContainer hidden">class="mathCode">$(p_1,p_2,p_3)\in P$ is adjacent to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si7.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=70929ec2285d0cd1851c380c8a47f845" title="Click to view the MathML source">[l₁,l₂,l₃]∈Lclass="mathContainer hidden">class="mathCode">$[l_1,l_2,l_3]\in L$ if and only if It is known that the graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si9.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=84468db723da330e43ede4744d5ac388" title="Click to view the MathML source">Γ_F(xy²)class="mathContainer hidden">class="mathCode">${\Gamma }_{\mathbb{F}}\left(x{y}^2\right)$ has no cycles of length less than eight. The main result of this paper is that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si9.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=84468db723da330e43ede4744d5ac388" title="Click to view the MathML source">Γ_F(xy²)class="mathContainer hidden">class="mathCode">${\Gamma }_{\mathbb{F}}\left(x{y}^2\right)$ is the only graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si3.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=2722c2becd8df07ccb26ceb9892ccba9" title="Click to view the MathML source">Γ_F(f)class="mathContainer hidden">class="mathCode">${\Gamma }_{\mathbb{F}}\left(f\right)$ with this property when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si1.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=f72988a8747a3b8bffaf5f249699d114" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">$\mathbb{F}$ is an algebraically closed field of characteristic zero; i.e. over such a field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si1.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=f72988a8747a3b8bffaf5f249699d114" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">$\mathbb{F}$, every graph class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si3.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=2722c2becd8df07ccb26ceb9892ccba9" title="Click to view the MathML source">Γ_F(f)class="mathContainer hidden">class="mathCode">${\Gamma }_{\mathbb{F}}\left(f\right)$ with no cycles of length less than eight is isomorphic to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si9.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=84468db723da330e43ede4744d5ac388" title="Click to view the MathML source">Γ_F(xy²)class="mathContainer hidden">class="mathCode">${\Gamma }_{\mathbb{F}}\left(x{y}^2\right)$. We also prove related uniqueness results for some polynomials class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X1630004X&_mathId=si16.gif&_user=111111111&_pii=S0166218X1630004X&_rdoc=1&_issn=0166218X&md5=97e12fb424fc5e6fd7e1c379e1e2b8ee" title="Click to view the MathML source">fclass="mathContainer hidden">class="mathCode">$f$ over infinite families of finite fields.