Edge-signed graphs with smallest eigenvalue greater than -2

Gary Greaves; Jack Koolen; Akihiro Munemasa; Yoshio Sano; Tetsuji Taniguchi

doi:10.1016/j.jctb.2014.07.006

Edge-signed graphs with smallest eigenvalue greater than -2

Gary Greaves, Jack Koolen, Akihiro Munemasa, Yoshio Sano, Tetsuji Taniguchi

INF - Computer and Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

10 Citations (Scopus)

Abstract

We give a structural classification of edge-signed graphs with smallest eigenvalue greater than -2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we prove a more general result extending Hoffman's original statement to all edge-signed graphs with smallest eigenvalue greater than -2. Our results give a classification of the special graphs of fat Hoffman graphs with smallest eigenvalue greater than -3.

Original language	English
Pages (from-to)	90-111
Number of pages	22
Journal	Journal of Combinatorial Theory. Series B
Volume	110
DOIs	https://doi.org/10.1016/j.jctb.2014.07.006
Publication status	Published - 2015 Jan 1

Keywords

Hoffman conjecture
Hoffman graphs
Line graphs
Signed graphs
Smallest eigenvalue

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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AU - Greaves, Gary

AU - Koolen, Jack

AU - Munemasa, Akihiro

AU - Sano, Yoshio

AU - Taniguchi, Tetsuji

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We give a structural classification of edge-signed graphs with smallest eigenvalue greater than -2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we prove a more general result extending Hoffman's original statement to all edge-signed graphs with smallest eigenvalue greater than -2. Our results give a classification of the special graphs of fat Hoffman graphs with smallest eigenvalue greater than -3.

AB - We give a structural classification of edge-signed graphs with smallest eigenvalue greater than -2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we prove a more general result extending Hoffman's original statement to all edge-signed graphs with smallest eigenvalue greater than -2. Our results give a classification of the special graphs of fat Hoffman graphs with smallest eigenvalue greater than -3.

KW - Hoffman conjecture

KW - Hoffman graphs

KW - Line graphs

KW - Signed graphs

KW - Smallest eigenvalue

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