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arXiv · 1409.5328

Bounds for the positive or negative inertia index of a graph

Yi-Zheng Fan·Long Wang
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Abstract

Let $G$ be a graph and let $A(G)$ be adjacency matrix of $G$.The positive inertia index (respectively, the negative inertia index) of $G$, denoted by $p(G)$ (respectively, $n(G)$), is defined to be the number of positive eigenvalues (respectively, negative eigenvalues) of $A(G)$. In this paper, we present the bounds for $p(G)$ and $n(G)$ as follows: $$m(G)-c(G)\leq p(G)\leq m(G)+c(G), \ m(G)-c(G)\leq n(G)\leq m(G)+c(G),$$ where $m(G)$ and $c(G)$ are respectively the matching number and the cyclomatic number of $G$. Furthermore, we characterize the graphs which attain the upper bounds or the lower bounds respectively.

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Yi-Zheng Fan, Long Wang. 2014-09-18. Bounds for the positive or negative inertia index of a graph. https://doi.org/10.1016/j.laa.2017.02.005

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