Geodesic
Revisiting two classical results on graph spectra
The electronic journal of combinatorics, Tome 14 (2007)

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Let $\mu\left( G\right) $ and $\mu_{\min}\left( G\right) $ 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 $$ \mu\left( G\right) -\mu\left( H\right) >{1\over\mu\left( G\right) ^{2D}n}. $$ (ii) If $G$ is a connected nonbipartite graph of order $n$ and diameter $D$, then $$ \mu\left( G\right) +\mu_{\min}\left( G\right) >{2\over\mu\left( G\right) ^{2D}n}. $$ For large $\mu $ and $D$ these bounds are close to the best possible ones.
DOI : 10.37236/932
Classification : 05C50 
Vladimir Nikiforov. Revisiting two classical results on graph spectra. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/932
@article{10_37236_932,
     author = {Vladimir Nikiforov},
     title = {Revisiting two classical results on graph spectra},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/932},
     zbl = {1111.05062},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/932/}
}
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