# On the Characteristic Polynomial and Eigenvectors in Terms of the Tree-Like Structure of a Digraph

## Abstract

Regarding a square matrix as the adjacency matrix of a weighted digraph, we construct an extended digraph whose Laplacian contains the original matrix as a submatrix. This construction allows us to use known results on Laplacians to study arbitrary square matrices. The calculation of an eigenvector in a parametric form demonstrates a connection between its components and the tree-like structure of the digraph.

- Authors:

- St.Petersburg State University (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22774017

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 232; Journal Issue: 1; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; EIGENVECTORS; GRAPH THEORY; LAPLACIAN; MATRICES; POLYNOMIALS

### Citation Formats

```
Buslov, V. A., E-mail: abvabv@bk.ru, E-mail: v.buslov@spbu.ru.
```*On the Characteristic Polynomial and Eigenvectors in Terms of the Tree-Like Structure of a Digraph*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3854-5.

```
Buslov, V. A., E-mail: abvabv@bk.ru, E-mail: v.buslov@spbu.ru.
```*On the Characteristic Polynomial and Eigenvectors in Terms of the Tree-Like Structure of a Digraph*. United States. doi:10.1007/S10958-018-3854-5.

```
Buslov, V. A., E-mail: abvabv@bk.ru, E-mail: v.buslov@spbu.ru. Sun .
"On the Characteristic Polynomial and Eigenvectors in Terms of the Tree-Like Structure of a Digraph". United States. doi:10.1007/S10958-018-3854-5.
```

```
@article{osti_22774017,
```

title = {On the Characteristic Polynomial and Eigenvectors in Terms of the Tree-Like Structure of a Digraph},

author = {Buslov, V. A., E-mail: abvabv@bk.ru, E-mail: v.buslov@spbu.ru},

abstractNote = {Regarding a square matrix as the adjacency matrix of a weighted digraph, we construct an extended digraph whose Laplacian contains the original matrix as a submatrix. This construction allows us to use known results on Laplacians to study arbitrary square matrices. The calculation of an eigenvector in a parametric form demonstrates a connection between its components and the tree-like structure of the digraph.},

doi = {10.1007/S10958-018-3854-5},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 1,

volume = 232,

place = {United States},

year = {2018},

month = {7}

}

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