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Uniquely Derived Generalized Line Graphs and Uniquely Represented Signed Graphs with Least Eigenvalues \(\ge \) \(-2\)

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Abstract

A well known result on line graphs due to H. Whitney is the following: Let \(\Omega \) be a (finite) connected line graph other than \(K_3\); then there is a unique connected graph \(G\) such that its line graph is \(\Omega \); further, if the order of \(G\) is more than \(4\), then the automorphism groups of \(G\) and \(\Omega \) are isomorphic. This result has been extended to (finite) generalized line graphs by P. J. Cameron. In this article, we derive two results which are related to Cameron’s result; one is on generalized line graphs and the other is on a class of signed graphs with least eigenvalues \({}\geqslant -2\). Digressing the course of obtaining the latter result, we prove a result on signed graphs which generalizes the following one due to A. Torga s̆ ev: If the least eigenvalue of a connected countably infinite graph is at least \(-2\), then it is a generalized line graph.

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Acknowledgments

The author is quite grateful to the referee for pointing out a number of minor mistakes, for stressing the need for adding more details in some of the proofs and for making some valuable suggestions to improve the presentation and the readability of this paper.

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

    G. R. Vijayakumar

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  1. G. R. Vijayakumar
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Vijayakumar, G.R. Uniquely Derived Generalized Line Graphs and Uniquely Represented Signed Graphs with Least Eigenvalues \(\ge \) \(-2\) . Graphs and Combinatorics 31, 1053–1063 (2015). https://doi.org/10.1007/s00373-014-1419-z

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