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A characterization of the smallest eigenvalue of a graph. (English) Zbl 0792.05096
Summary: It is well known that the smallest eigenvalue of the adjacency matrix of a connected $$d$$-regular graph is at least $$-d$$ and is strictly greater than $$-d$$ if the graph is not bipartite. More generally, for any connected graph $$G=(V,E)$$, consider the matrix $$Q=D+A$$ where $$D$$ is the diagonal matrix of degrees in the graph $$G$$ and $$A$$ is the adjacency matrix of $$G$$. Then $$Q$$ is positive semidefinite, and the smallest eigenvalue of $$Q$$ is 0 if and only if $$G$$ is bipartite. We will study the separation of this eigenvalue from 0 in terms of the following measure of nonbipartiteness of $$G$$. For any $$S \subseteq V$$, we denote by $$e_{\min} (S)$$ the minimum number of edges that need to be removed from the induced subgraph on $$S$$ to make it bipartite. Also, we denote by $$\text{cut} (S)$$ the set of edges with one end in $$S$$ and the other in $$V-S$$. We define the parameter $$\psi$$ as $\psi=\min_{S \subseteq V} {e_{\min} (S)+| \text{cut} (S) | \over | S |}.$ The parameter $$\psi$$ is a measure of the nonbipartiteness of the graph $$G$$. We will show that the smallest eigenvalue of $$Q$$ is bounded above and below by functions of $$\psi$$. For $$d$$-regular graphs, this characterizes the separation of the smallest eigenvalue of the adjacency matrix from $$- d$$. These results can be easily extended to weighted graphs.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
##### Keywords:
$$d$$-regular graph; smallest eigenvalue; adjacency matrix
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