Characterizations of the family of all generalized line graphs-finite and infinite- and classification of the family of all graphs whose least eigenvalues \ge - 2

Abstract

The infimum of the least eigenvalues of all finite induced subgraphs of an infinite graph is defined to be its least eigenvalue. In [P.J. Cameron, J.M. Goethals, J.J. Seidel and E.E. Shult, Line graphs, root systems, and elliptic geometry, J. Algebra 43 (1976) 305-327], the class of all finite graphs whose least eigenvalues −2 has been classified: (1) If a (finite) graph is connected and its least eigenvalue is at least −2, then either it is a generalized line graph or it is represented by the root system E 8 . In [A. Torgasev, A note on infinite generalized line graphs, in: Proceedings of the Fourth Yugoslav Seminar on Graph Theory, Novi Sad, 1983 (Univ. Novi Sad, 1984, it has been found that (2) any countably infinite connected graph with least eigenvalue −2 is a generalized line graph. In this article, the family of all generalized line graphs-countable and uncountable-is described algebraically and characterized structurally and an extension of (1) which subsumes (2) is derived.