Editing Spectral radius (section)

==Upper bounds==

===Upper bounds on the spectral radius of a matrix===

The following proposition gives simple yet useful upper bounds on the spectral radius of a matrix.

'''Proposition.''' Let {{math|''A'' ∈ '''C'''<sup>''n''×''n''</sup>}} with spectral radius {{math|''ρ''(''A'')}} and a [[sub-multiplicative norm|sub-multiplicative matrix norm]] {{math|{{!!}}⋅{{!!}}}}. Then for each integer <math>k \geqslant 1</math>:

::<math>\rho(A)\leq \|A^k\|^{\frac{1}{k}}.</math>

'''Proof'''

Let {{math|('''v''', ''λ'')}} be an [[eigenvector]]-[[eigenvalue]] pair for a matrix ''A''. By the sub-multiplicativity of the matrix norm, we get:

:<math>|\lambda|^k\|\mathbf{v}\| = \|\lambda^k \mathbf{v}\| = \|A^k \mathbf{v}\| \leq \|A^k\|\cdot\|\mathbf{v}\|.</math>

Since {{math|'''v''' ≠ 0}}, we have

:<math>|\lambda|^k \leq \|A^k\|</math>

and therefore

:<math>\rho(A)\leq \|A^k\|^{\frac{1}{k}}.</math>

concluding the proof.

=== Upper bounds for spectral radius of a graph ===
There are many upper bounds for the spectral radius of a graph in terms of its number ''n'' of vertices and its number ''m'' of edges. For instance, if

:<math>\frac{(k-2)(k-3)}{2} \leq m-n \leq \frac{k(k-3)}{2}</math>

where <math>3 \le k \le n</math> is an integer, then<ref>{{Cite journal|last1=Guo|first1=Ji-Ming|last2=Wang|first2=Zhi-Wen|last3=Li|first3=Xin|date=2019|title=Sharp upper bounds of the spectral radius of a graph|journal=Discrete Mathematics|language=en|volume=342|issue=9|pages=2559–2563|doi=10.1016/j.disc.2019.05.017|s2cid=198169497|doi-access=free}}</ref>

:<math>\rho(G) \leq \sqrt{2 m-n-k+\frac{5}{2}+\sqrt{2 m-2 n+\frac{9}{4}}}</math>