# Bounds for index of a modified graph

• Volume: 24, Issue: 2, page 213-221
• ISSN: 2083-5892

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## Abstract

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If a graph is connected then the largest eigenvalue (i.e., index) generally changes (decreases or increases) if some local modifications are performed. In this paper two types of modifications are considered: (i) for a fixed vertex, t edges incident with it are deleted, while s new edges incident with it are inserted; (ii) for two non-adjacent vertices, t edges incident with one vertex are deleted, while s new edges incident with the other vertex are inserted. Within each case, we provide lower and upper bounds for the indices of the modified graphs, and then give some sufficient conditions for the index to decrease or increase when a graph is modified as above.

## How to cite

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Bo Zhou. "Bounds for index of a modified graph." Discussiones Mathematicae Graph Theory 24.2 (2004): 213-221. <http://eudml.org/doc/270543>.

@article{BoZhou2004,
abstract = { If a graph is connected then the largest eigenvalue (i.e., index) generally changes (decreases or increases) if some local modifications are performed. In this paper two types of modifications are considered: (i) for a fixed vertex, t edges incident with it are deleted, while s new edges incident with it are inserted; (ii) for two non-adjacent vertices, t edges incident with one vertex are deleted, while s new edges incident with the other vertex are inserted. Within each case, we provide lower and upper bounds for the indices of the modified graphs, and then give some sufficient conditions for the index to decrease or increase when a graph is modified as above. },
author = {Bo Zhou},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph; eigenvalue; principal eigenvector},
language = {eng},
number = {2},
pages = {213-221},
title = {Bounds for index of a modified graph},
url = {http://eudml.org/doc/270543},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Bo Zhou
TI - Bounds for index of a modified graph
JO - Discussiones Mathematicae Graph Theory
PY - 2004
VL - 24
IS - 2
SP - 213
EP - 221
AB - If a graph is connected then the largest eigenvalue (i.e., index) generally changes (decreases or increases) if some local modifications are performed. In this paper two types of modifications are considered: (i) for a fixed vertex, t edges incident with it are deleted, while s new edges incident with it are inserted; (ii) for two non-adjacent vertices, t edges incident with one vertex are deleted, while s new edges incident with the other vertex are inserted. Within each case, we provide lower and upper bounds for the indices of the modified graphs, and then give some sufficient conditions for the index to decrease or increase when a graph is modified as above.
LA - eng
KW - graph; eigenvalue; principal eigenvector
UR - http://eudml.org/doc/270543
ER -

## References

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1. [1] D. Cvetković, P. Rowlinson and S. Simić, Eigenspaces of graphs (Cambridge University Press, Cambridge, 1997). Zbl0878.05057
2. [2] C. Maas, Perturbation results for adjacency spectrum of a graph, Z. Angew. Math. Mech. 67 (1987) 428-430.
3. [3] P. Rowlinson, On angles and perturbations of graphs, Bull. London Math. Soc. 20 (1988) 193-197, doi: 10.1112/blms/20.3.193. Zbl0644.05038
4. [4] P. Rowlinson, More on graph perturbations, Bull. London Math. Soc. 22 (1990) 209-216, doi: 10.1112/blms/22.3.209. Zbl0711.05034
5. [5] W. Weinstein and W. Stenger, Methods of intermediate problems of eigenvalues (Academic Press, New York, 1972). Zbl0291.49034
6. [6] B. Zhou, The changes in indices of modified graphs, Linear Algebra Appl. 356 (2002) 95-101, doi: 10.1016/S0024-3795(02)00321-X. Zbl1015.05046

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