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An inequality involving the second largest and smallest eigenvalue of a distance-regular graph
Koolen, Jack H.,Park, Jongyook,Yu, Hyonju Elsevier 2011 Linear algebra and its applications Vol.434 No.12
<P><B>Abstract</B></P><P>For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) <SUB>θ1</SUB> (resp., <SUB>θD</SUB>) we show that (<SUB>θ1</SUB>+1)(<SUB>θD</SUB>+1)⩽-<SUB>b1</SUB> holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.</P>
The non-bipartite integral graphs with spectral radius three
Chung, Taeyoung,Koolen, Jack,Sano, Yoshio,Taniguchi, Tetsuji Elsevier 2011 Linear algebra and its applications Vol.435 No.10
<P><B>Abstract</B></P><P>In this paper, we classify the connected non-bipartite integral graphs with spectral radius three.</P>
Partially metric association schemes with a multiplicity three
van Dam, Edwin R.,Koolen, Jack H.,Park, Jongyook Elsevier 2018 Journal of combinatorial theory. Series B Vol.130 No.-
<P><B>Abstract</B></P> <P>An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete 4-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known 2-arc-transitive covers of the cube: the Möbius–Kantor graph, the Nauru graph, and the Foster graph F048A. In order to obtain this result, we also determine the symmetric association schemes with a multiplicity three and a connected relation with valency three. Moreover, we construct an infinite family of cubic arc-transitive 2-walk-regular graphs with an eigenvalue with multiplicity three that give rise to non-commutative association schemes with a symmetric relation of valency three and an eigenvalue with multiplicity three.</P>