论文标题
无限图的光谱具有可总结重量功能的光谱
Spectra of infinite graphs with summable weight functions
论文作者
论文摘要
在本文中，我们研究了无限加权图的laplacians的光谱。而不是局部有限的假设，而是强加了重量函数的总结性。这样的图对应于具有可数状态空间的可逆马尔可夫链。我们在这种环境中采用了cheeger常数的概念，并证明了表征光谱差距的Cheeger不平等的类似物。我们还分析了最初在\ cite {b14}中引入的双cheeger常数的概念，该常量{b14}允许估计频谱的顶部。在本文中，我们还引入了一种新的组合不变的K $（g，m）$，它允许对两部分图进行完整表征，并测量光谱的不对称性（频谱之间的Hausdorff距离及其在$ 1 \ in \ bb r r $ 1 \ in \ in \ bb r $之间的距离之间的距离）。我们将K $（G，M）$与Cheeger和双脸颊常数进行比较。最后，我们完整详细分析了一类无限完整的图形及其光谱。
In this paper we study spectra of Laplacians of infinite weighted graphs. Instead of the assumption of local finiteness we impose the condition of summability of the weight function. Such graphs correspond to reversible Markov chains with countable state spaces. We adopt the concept of the Cheeger constant to this setting and prove an analogue of the Cheeger inequality characterising the spectral gap. We also analyse the concept of the dual Cheeger constant originally introduced in \cite{B14}, which allows estimating the top of the spectrum. In this paper we also introduce a new combinatorial invariant, k$(G,m)$, which allows a complete characterisation of bipartite graphs and measures the asymmetry of the spectrum (the Hausdorff distance between the spectrum and its reflection at point $1\in \Bbb R$). We compare k$(G, m)$ to the Cheeger and the dual Cheeger constants. Finally, we analyse in full detail a class of infinite complete graphs and their spectra.
