# The inertia of unicyclic graphs and bicyclic graphs

• Volume: 33, Issue: 1, page 109-115
• ISSN: 1509-9415

top

## Abstract

top
Let G be a graph with n vertices and ν(G) be the matching number of G. The inertia of a graph G, In(G) = (n₊,n₋,n₀) is an integer triple specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix A(G), respectively. Let η(G) = n₀ denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G) = n - 2ν(G). Guo et al. [Ji-Ming Guo, Weigen Yan and Yeong-Nan Yeh. On the nullity and the matching number of unicyclic graphs, Linear Algebra and its Applications, 431 (2009), 1293-1301.] proved if G is a unicyclic graph, then η(G) equals n - 2ν(G) - 1, n-2ν(G) or n - 2ν(G) + 2. Barrett et al. determined the inertia sets for trees and graphs with cut vertices. In this paper, we give the nullity of bicyclic graphs 𝓑ₙ⁺⁺. Furthermore, we determine the inertia set in unicyclic graphs and 𝓑ₙ⁺⁺, respectively.

## How to cite

top

Ying Liu. "The inertia of unicyclic graphs and bicyclic graphs." Discussiones Mathematicae - General Algebra and Applications 33.1 (2013): 109-115. <http://eudml.org/doc/270634>.

@article{YingLiu2013,
abstract = {Let G be a graph with n vertices and ν(G) be the matching number of G. The inertia of a graph G, In(G) = (n₊,n₋,n₀) is an integer triple specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix A(G), respectively. Let η(G) = n₀ denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G) = n - 2ν(G). Guo et al. [Ji-Ming Guo, Weigen Yan and Yeong-Nan Yeh. On the nullity and the matching number of unicyclic graphs, Linear Algebra and its Applications, 431 (2009), 1293-1301.] proved if G is a unicyclic graph, then η(G) equals n - 2ν(G) - 1, n-2ν(G) or n - 2ν(G) + 2. Barrett et al. determined the inertia sets for trees and graphs with cut vertices. In this paper, we give the nullity of bicyclic graphs 𝓑ₙ⁺⁺. Furthermore, we determine the inertia set in unicyclic graphs and 𝓑ₙ⁺⁺, respectively.},
author = {Ying Liu},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {matching number; inertia; nullity; unicyclic graph; bicyclic graph},
language = {eng},
number = {1},
pages = {109-115},
title = {The inertia of unicyclic graphs and bicyclic graphs},
url = {http://eudml.org/doc/270634},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Ying Liu
TI - The inertia of unicyclic graphs and bicyclic graphs
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2013
VL - 33
IS - 1
SP - 109
EP - 115
AB - Let G be a graph with n vertices and ν(G) be the matching number of G. The inertia of a graph G, In(G) = (n₊,n₋,n₀) is an integer triple specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix A(G), respectively. Let η(G) = n₀ denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G) = n - 2ν(G). Guo et al. [Ji-Ming Guo, Weigen Yan and Yeong-Nan Yeh. On the nullity and the matching number of unicyclic graphs, Linear Algebra and its Applications, 431 (2009), 1293-1301.] proved if G is a unicyclic graph, then η(G) equals n - 2ν(G) - 1, n-2ν(G) or n - 2ν(G) + 2. Barrett et al. determined the inertia sets for trees and graphs with cut vertices. In this paper, we give the nullity of bicyclic graphs 𝓑ₙ⁺⁺. Furthermore, we determine the inertia set in unicyclic graphs and 𝓑ₙ⁺⁺, respectively.
LA - eng
KW - matching number; inertia; nullity; unicyclic graph; bicyclic graph
UR - http://eudml.org/doc/270634
ER -

## References

top
1. [1] W. Barrett, H. Tracy Hall and R. Loewy, The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample, Linear Algebra and its Applications 431 (2009) 1147-1191. doi: 10.1016/j.laa.2009.04.007. Zbl1175.05032
2. [2] D. Cvetkociić, M. Doob and H. Sachs, Spectra of Graphs - Theory and Application (Academic Press, New York, 1980).
3. [3] D. Cvetkocić, I. Gutman and N. Trinajstić, Graph theory and molecular orbitals II, Croat.Chem. Acta 44 (1972) 365-374.
4. [4] S. Fiorini, I. Gutman and I. Sciriha, Trees with maximum nullity, Linear Algebra and its Applications 397 (2005) 245-252. doi: 10.1016/j.laa.2004.10.024.
5. [5] Ji-Ming Guo, Weigen Yan and Yeong-Nan Yeh, On the nullity and the matching number of unicyclic graphs, Linear Algebra and its Applications 431 (2009) 1293-1301. doi: 10.1016/j.laa.2009.04.026. Zbl1238.05160
6. [6] Shengbiao Hu, Tan Xuezhong and Bolian Liu, On the nullity of bicyclic graphs, Linear Algebra and its Applications 429 (2008) 1387-1391. doi: 10.1016/j.laa.2007.12.007. Zbl1144.05319

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.