# The inertia of unicyclic graphs and bicyclic graphs

Discussiones Mathematicae - General Algebra and Applications (2013)

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

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topYing 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] 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] D. Cvetkociić, M. Doob and H. Sachs, Spectra of Graphs - Theory and Application (Academic Press, New York, 1980).
- [3] D. Cvetkocić, I. Gutman and N. Trinajstić, Graph theory and molecular orbitals II, Croat.Chem. Acta 44 (1972) 365-374.
- [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] 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] 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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