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 
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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/