On graphs with exactly two positive eigenvalues

Fang Duan, Qiongxiang Huang, Xueyi Huang

Abstract


The inertia of a graph G is defined to be the triplet In(G) = (p(G), n(G), η(G)), where p(G), n(G) and η(G) are the numbers of positive, negative and zero eigenvalues (including multiplicities) of the adjacency matrix A(G), respectively. Traditionally p(G) (resp. n(G)) is called the positive (resp. negative) inertia index of G. In this paper, we introduce three types of congruent transformations for graphs that keep the positive inertia index and negative inertia index. By using these congruent transformations, we determine all graphs with exactly two positive eigenvalues and one zero eigenvalue.


Keywords


Congruent transformation, positive (negative) inertia index, nullity

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DOI: https://doi.org/10.26493/1855-3974.1516.58f

ISSN: 1855-3974

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