Geodesic
The spectral radius of subgraphs of regular graphs
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) Let $G$ be a regular graph of order $n$ and finite diameter $D.$ If $H$ is a proper subgraph of $G,$ then $$ \mu\left( G\right) -\mu\left( H\right) >{1\over nD}. $$ (ii) If $G$ is a regular nonbipartite graph of order $n$ and finite diameter $D$, then $$ \mu\left( G\right) +\mu_{\min}\left( G\right) >{1\over nD}. $$
DOI : 10.37236/1021
Classification : 05C50
Mots-clés : smallest eigenvalue, largest eigenvalue, diameter, connected graph 
Vladimir Nikiforov. The spectral radius of subgraphs of regular graphs. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/1021
@article{10_37236_1021,
     author = {Vladimir Nikiforov},
     title = {The spectral radius of subgraphs of regular graphs},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/1021},
     zbl = {1157.05313},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1021/}
}
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