NDLI: Uniquely Derived Generalized Line Graphs and Uniquely Represented Signed Graphs with Least Eigenvalues $$\ge $$ $$-2$$

Content Provider	Springer Nature Link
Author	Vijayakumar, G. R.
Copyright Year	2014
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.
Ending Page	1063
Page Count	11
Starting Page	1053
File Format	PDF
ISSN	09110119
e-ISSN	14355914
Journal	Graphs and Combinatorics
Issue Number	4
Volume Number	31
Language	English
Publisher	Springer Japan
Publisher Date	2014-04-20
Publisher Place	Tokyo
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Innergraph Uniquely derived graph Signed and weighted graphs Least eigenvalue of a sigraph Generalized line graph Structural characterization of families of graphs Representation of a sigraph Infinite graphs Engineering Design Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science

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