The inertia of unicyclic graphs and bicyclic graphs

Ying Liu

Discussiones Mathematicae - General Algebra and Applications (2013)

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

Abstract

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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

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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

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  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

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