Geodesic
Characterizations of the Family of All Generalized Line Graphs—Finite and Infinite—and Classification of the Family of All Graphs Whose Least Eigenvalues ≥ −2
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 637-648

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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 E8. In [A. Torgašev, A note on infinite generalized line graphs, in: Proceedings of the Fourth Yugoslav Seminar on Graph Theory, Novi Sad, 1983 (Univ. Novi Sad, 1984) 291- 297], 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.
Keywords: generalized line graph, enhanced line graph, representation of a graph, extended line graph, least eigenvalue of a graph 
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Vijayakumar, Gurusamy Rengasamy. Characterizations of the Family of All Generalized Line Graphs—Finite and Infinite—and Classification of the Family of All Graphs Whose Least Eigenvalues ≥ −2. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 637-648. http://geodesic.mathdoc.fr/item/DMGT_2013_33_4_a0/