## Revisiting two classical results on graph spectra.(English)Zbl 1111.05062

Summary: Let $$\mu(G)$$ and $$\mu_{\min}(G)$$ be the largest and smallest eigenvalues of the adjacency matrix of a graph $$G$$. Our main results are: (i) if $$H$$ is a proper subgraph of a connected graph $$G$$ of order $$n$$ and diameter $$D$$, then $$\mu(G) -\mu(H) > 1/(\mu(G)^{2D}n)$$, and (ii) if $$G$$ is a connected nonbipartite graph of order $$n$$ and diameter $$D$$, then $$\mu(G) +\mu_{\min}(G) > 2/(\mu(G)^{2D}n).$$ For large $$\mu$$ and $$D$$ these bounds are close to the best possible ones.

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text: