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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1196,
author = {Ying Liu},
title = {The inertia of unicyclic graphs and bicyclic graphs},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {33},
year = {2013},
pages = {109-115},
zbl = {1293.05213},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1196}
}
Ying Liu. The inertia of unicyclic graphs and bicyclic graphs. Discussiones Mathematicae - General Algebra and Applications, Tome 33 (2013) pp. 109-115. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1196/
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