Extremal inverse eigenvalue problem for matrices described by a connected unicyclic graph

Bijoya Bardhan; Mausumi Sen; Debashish Sharma

Applications of Mathematics (2024)

  • Volume: 69, Issue: 2, page 273-286
  • ISSN: 0862-7940

Abstract

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In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation.

How to cite

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Bardhan, Bijoya, Sen, Mausumi, and Sharma, Debashish. "Extremal inverse eigenvalue problem for matrices described by a connected unicyclic graph." Applications of Mathematics 69.2 (2024): 273-286. <http://eudml.org/doc/299256>.

@article{Bardhan2024,
abstract = {In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation.},
author = {Bardhan, Bijoya, Sen, Mausumi, Sharma, Debashish},
journal = {Applications of Mathematics},
keywords = {inverse eigenvalue problem; unicyclic graph; leading principal submatrices},
language = {eng},
number = {2},
pages = {273-286},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Extremal inverse eigenvalue problem for matrices described by a connected unicyclic graph},
url = {http://eudml.org/doc/299256},
volume = {69},
year = {2024},
}

TY - JOUR
AU - Bardhan, Bijoya
AU - Sen, Mausumi
AU - Sharma, Debashish
TI - Extremal inverse eigenvalue problem for matrices described by a connected unicyclic graph
JO - Applications of Mathematics
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 2
SP - 273
EP - 286
AB - In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation.
LA - eng
KW - inverse eigenvalue problem; unicyclic graph; leading principal submatrices
UR - http://eudml.org/doc/299256
ER -

References

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