A Local Approach to 1-Homogeneous Graphs |
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Authors: | Aleksandar Jurivsic Jack Koolen |
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Affiliation: | (1) Nova Gorica Polytechnic, Vipavska 13, p.p. 301 Nova Gorica, Slovenia;(2) Centrum voor Wiskunde en Informatica, Kruislaan, 413, NL-1098SJ Amsterdam, The Netherlands |
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Abstract: | Let bea distance-regular graph with diameter d. For vertices x and y of at distancei, 1 i d, we define the setsCi(x,y) = i–1(x) (y), Ai(x,y) = i(x) (y) and Bi(x,y) = i+1(x) (y).Then we say has the CABj property,if the partition CABi(x,y) = {Ci(x,y),Ai(x,y),Bi(x,y)}of the local graph of y is equitable for each pairof vertices x and y of at distance i j. We show that in with the CABj property then the parameters ofthe equitable partitions CABi(x,y) do not dependon the choice of vertices x and y atdistance i for all i j. The graph has the CAB property if it has the CABdproperty. We show the equivalence of the CAB property and the1-homogeneous property in a distance-regular graph with a1 0. Finally, we classify the 1-homogeneous Terwilligergraphs with c2 2. |
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Keywords: | Distance-regular graphs equitable partitions 1-homogeneous locally strongly-regular |
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