The inertia of unicyclic graphs and bicyclic graphs

• Autorzy: Ying Liu
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

matching number; inertia; nullity; unicyclic graph; bicyclic graph

Kategorie tematyczne

• 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2013
• Tom: 33
• Numer: 1
• Strony: 109-115

Daty

wydano

2013

otrzymano

2013-03-08

poprawiono

2013-03-26

Twórcy

autor

Ying Liu

• School of Mathematics and Information, Shanghai Lixin University of Commerce, Shanghai, 201620, China

Bibliografia

• [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.
• [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.
• [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.

Identyfikatory

DOI

10.7151/dmgaa.1196