Star complements and exceptional graphs

Let G be a finite graph of order n with an eigenvalue μ of multiplicity k. (Thus the μ-eigenspace of a (0, 1)-adjacency matrix of G has dimension k.) Astar complement for μ in G is an induced subgraph G − X of G such that |X |= k and G − X does not have μ as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, whose eigenvalues lie in [−2, ∞). We establish some properties of star complements, and of eigenvectors, of exceptional graphs with least eigenvalue −2.

[1]  D. Cvetkovic,et al.  COSPECTRAL GRAPHS WITH LEAST EIGENVALUE AT LEAST ¡2 , 2005 .

[2]  D. Cvetkovic,et al.  Eigenspaces of graphs: Bibliography , 1997 .

[3]  Michael Doob,et al.  On spectral characterizations and embeddings of graphs , 1979 .

[4]  A. Hora,et al.  Distance-Regular Graphs , 2007 .

[5]  Peter Rowlinson,et al.  Co-cliques and star complements in extremal strongly regular graphs , 2007 .

[6]  Mark N. Ellingham,et al.  Basic subgraphs and graph spectra , 1993, Australas. J Comb..

[7]  Dragoš Cvetković,et al.  Spectral Generalizations of Line Graphs: On Graphs with Least Eigenvalue -2 , 2004 .

[8]  Peter Rowlinson,et al.  On the Multiplicities of Graph Eigenvalues , 2003 .

[9]  Michael Doob,et al.  Spectra of graphs , 1980 .

[10]  P. Rowlinson,et al.  COMPUTER INVESTIGATIONS OF THE MAXIMAL EXCEPTIONAL GRAPHS , 2001 .

[11]  Elias M. Hagos.  Some results on graph spectra , 2002 .

[12]  A. Neumaier,et al.  Exceptional graphs with smallest eigenvalue -2 and related problems , 1992 .

[13]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[14]  D. Cvetkovic,et al.  Graphs with Least Eigenvalue −2: The Star Complement Technique , 2001 .

[15]  A. Mukherjee,et al.  Two new graph-theoretical methods for generation of eigenvectors of chemical graphs , 1989 .

[16]  Dragoš Cvetković,et al.  Spectral Generalizations of Line Graphs: Preface , 2004 .

[17]  Mirko Lepovic,et al.  The Maximal Exceptional Graphs , 2002, J. Comb. Theory B.

[18]  D. Cvetkovic.  Constructions of the Maximal Exceptional Graphs with Largest Degree Less than 28 , 2000 .