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Eigenvalues and domination in graphs. (English) Zbl 0857.05072
For a finite connected graph $$G$$ with $$n$$ vertices let $$\gamma(G)$$ be the domination number and $$\lambda_n$$ the largest eigenvalue of its Laplacian matrix. The authors then establish the following results: (i) If $$\gamma(G)\geq 3$$, then $$\lambda_n< n-\lceil{\gamma(G)-2\over 2}\rceil$$. (ii) If $$\gamma(G)= 1$$, then $$\lambda_n=n$$. (iii) If $$\gamma(G)=2$$, no better bound than $$\lambda_n\leq n$$ exists. (iv) Eigenvectors corresponding to large eigenvalues induce dominating sets in $$G$$.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C35 Extremal problems in graph theory
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##### References:
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