Geodesic
The inertia of unicyclic graphs and bicyclic graphs
Discussiones Mathematicae. General Algebra and Applications, Tome 33 (2013) no. 1, pp. 109-115

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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 ₙ⁺⁺. Furthermore, we determine the inertia set in unicyclic graphs and ₙ⁺⁺, respectively.
Keywords: matching number, inertia, nullity, unicyclic graph, bicyclic graph 
Liu, Ying. The inertia of unicyclic graphs and bicyclic graphs. Discussiones Mathematicae. General Algebra and Applications, Tome 33 (2013) no. 1, pp. 109-115. http://geodesic.mathdoc.fr/item/DMGAA_2013_33_1_a7/
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