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Shilla distance-regular graphs
Koolen, J.H.,Park, J. Academic Press 2010 European journal of combinatorics : Journal europ& Vol.31 No.8
A Shilla distance-regular graph Γ (say with valency k) is a distance-regular graph with diameter 3 such that its second-largest eigenvalue equals a<SUB>3</SUB>. We will show that a<SUB>3</SUB> divides k for a Shilla distance-regular graph Γ, and for Γ we define b=b(Γ)@?ka<SUB>3</SUB>. In this paper we will show that there are finitely many Shilla distance-regular graphs Γ with fixed b(Γ)>=2. Also, we will classify Shilla distance-regular graphs with b(Γ)=2 and b(Γ)=3. Furthermore, we will give a new existence condition for distance-regular graphs, in general.
Injective optimal realizations of finite metric spaces
Koolen, J.H.,Lesser, A.,Moulton, V.,Wu, T. North-Holland Pub. Co ; Elsevier Science Ltd 2012 Discrete mathematics Vol.312 No.10
A realization of a finite metric space (X,d) is a weighted graph (G,w) whose vertex set contains X such that the distances between the elements of X in G correspond to those given by d. Such a realization is called optimal if it has minimal total edge weight. Optimal realizations have applications in fields such as phylogenetics, psychology, compression software and internet tomography. Given an optimal realization (G,w) of (X,d), there always exist certain ''proper'' maps from the vertex set of G into the so-called tight span of d. In [A. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. Math. 53 (1984) 321-402], Dress conjectured that any such map must be injective. Although this conjecture was recently disproven, in this paper we show that it is possible to characterize those optimal realizations (G,w) for which certain generalizations of proper maps-that map the geometric realization of (G,w) into the tight span instead of its vertex set-must always be injective. We also prove that these ''injective'' optimal realizations always exist, and show how they may be constructed from non-injective ones. Ultimately it is hoped that these results will contribute towards developing new ways to compute optimal realizations from tight spans.
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>
Improving diameter bounds for distance-regular graphs
Bang, S.,Hiraki, A.,Koolen, J.H. Elsevier 2006 European journal of combinatorics : Journal europ& Vol.27 No.1
<P><B>Abstract</B></P><P>In this paper we study the sequence <SUB>(<SUB>ci</SUB>)0≤i≤d</SUB> for a distance-regular graph. In particular we show that if d≥2j and <SUB>cj</SUB>>1 then <SUB>c2j−1</SUB>><SUB>cj</SUB> holds. Using this we give improvements on diameter bounds by A. Hiraki, J.H. Koolen [An improvement of the Ivanov bound, Ann. Comb. 2 (2) (1998) 131–135], and L. Pyber [A bound for the diameter of distance-regular graphs, Combinatorica 19 (4) (1999) 549–553], respectively, by applying this inequality.</P>
2-Walk-regular graphs with a small number of vertices compared to the valency
Qiao, Z.,Koolen, J.H.,Park, J. North Holland [etc.] 2016 Linear algebra and its applications Vol.510 No.-
<P>In 2013, it was shown that, for a given real number alpha > 2, there are only finitely many distance-regular graphs Gamma with valency k and diameter D >= 3 having at most alpha k vertices, except for the following two cases: (i) D = 3 and Gamma is imprimitive; (ii) D = 4 and Gamma 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. (C) 2016 Elsevier Inc. All rights reserved.</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>