2-Walk-regular graphs with a small number of vertices compared to the valency

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• 作者：Zhi Qiao^a ; ^{zhiqiao@sicnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Jack H. Koolen^b ; ^c ; ^{koolen@ustc.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Jongyook Park^d ; ^{jongyook@hanmail.net" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05E30 ; 05C50
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：510
• 期：Complete
• 页码：10-24
• 全文大小：415 K

文摘

In 2013, it was shown that, for a given real number an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516303275&_mathId=si1.gif&_user=111111111&_pii=S0024379516303275&_rdoc=1&_issn=00243795&md5=5cdc1b91755a2b33e0d05c84d7e727c5" title="Click to view the MathML source">α>2an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">α>2ath>an>an>an>, there are only finitely many distance-regular graphs Γ with valency k   and diameter an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516303275&_mathId=si2.gif&_user=111111111&_pii=S0024379516303275&_rdoc=1&_issn=00243795&md5=0c924384d9868ce57c158c74e5bdaf53" title="Click to view the MathML source">D≥3an>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">D≥3ath>an>an>an> having at most αk vertices, except for the following two cases: (i  ) an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516303275&_mathId=si3.gif&_user=111111111&_pii=S0024379516303275&_rdoc=1&_issn=00243795&md5=5cd38161d9a7879b87b59afceceb92ec" title="Click to view the MathML source">D=3an>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">D=3ath>an>an>an> and Γ is imprimitive; an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516303275&_mathId=si4.gif&_user=111111111&_pii=S0024379516303275&_rdoc=1&_issn=00243795&md5=e880ec2559a63dd0cd77df9c017003d7" title="Click to view the MathML source">(ii)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">alse">(iialse">)ath>an>an>an>an id="mmlsi10" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516303275&_mathId=si10.gif&_user=111111111&_pii=S0024379516303275&_rdoc=1&_issn=00243795&md5=13ab34c75db013124e67c06eea478c27" title="Click to view the MathML source">D=4an>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">D=4ath>an>an>an> and Γ is antipodal and bipartite. In this paper, we will generalize this result to 2-walk-regular graphs. In this case, also incidence graphs of certain group divisible designs appear.