Discussiones Mathematicae Graph Theory 33(4) (2013) 637-648
doi: 10.7151/dmgt.1691

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Characterizations of the Family of All Generalized Line Graphs---Finite and Infinite---and Classification of the Family of All Graphs Whose Least Eigenvalues ≥-2

Gurusamy Rengasamy Vijayakumar

School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road, Colaba, Mumbai 400,005, India

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 E8. 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) 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

2010 Mathematics Subject Classification: 05C75, 05C63.

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Received 28 December 2011
Revised 15 July 2012
Accepted 16 July 2012