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, p. 637.

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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.
Classification : 05C50, 05C63, 05C75, 05C76
Mots-clés : generalized line graph, enhanced line graph, representation of a graph, extended line graph, least eigenvalue of a graph 
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     title = {Characterizations of the {Family} of {All} {Generalized} {Line} {Graphs-Finite} and {Infinite-and} {Classification} of the {Family} of {All} {Graphs} {Whose} {Least} {Eigenvalues} \ensuremath{\geq} \ensuremath{-}2},
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Gurusamy Rengasamy Vijayakumar. 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, p. 637. https://geodesic-test.mathdoc.fr/item/DMGT_2013__33_4_267901/