论文标题
在本地有限图的邻接运营商上
On adjacency operators of locally finite graphs
论文作者
论文摘要
如果对于$γ$ $γ$的每个顶点$ v $，则图$γ$称为本地有限的，而所有$ v $ $ v $ in $γ$的$γ（v）$是有限的。对于任何本地有限的图形$γ$，带有顶点套装$ v（γ）$，对于任何字段$ f $，让$ f^{v（γ）} $是所有功能$ v（γ）\ to f $ y to f $ y to f $的矢量空格$ f^{v（γ）} \ to f^{v（γ）} $由$（a^{（{{\ rm alg}）} _ {γ，f}（f）（f）（v）= \ sum_ {u sum_ {u sum_ {u s inγ（v）} f（v）f（v）f（v） V（γ）$。在有限图的情况下，映射$ a^{（（{\ rm alg}）} _ {γ，f} $是$γ$（超过$ f $）的邻接矩阵定义的众所周知的操作员，并且是eigenvaluese and eigenfunctions of eigenvaluese and eigenfunctions of the Opterator f. \ Mathbb {C} $）有限图理论的一部分。在本文中，我们取消了$ a^{（{{{\ rm alg}）} _ {γ，f} $的特征值和特征功能的理论，用于本地无限的图形$γ$（尽管有几个结果可能是有限的$ f $ fields $ f $ fields $ f $）学位和$ f = \ mathbb {c} $。根据作者的意见，以前朝这个方向的尝试并不十分令人满意，因为考虑到相当特殊的特征函数和相应的特征值受到限制。
A graph $Γ$ is called locally finite if, for each vertex $v$ of $Γ$, the set $Γ(v)$ of all neighbors of $v$ in $Γ$ is finite. For any locally finite graph $Γ$ with vertex set $V(Γ)$ and for any field $F$, let $F^{V(Γ)}$ be the vector space over $F$ of all functions $V(Γ) \to F$ (with natural componentwise operations) and let $A^{({\rm alg})}_{Γ,F}$ be the linear operator $F^{V(Γ)} \to F^{V(Γ)}$ defined by $(A^{({\rm alg})}_{Γ,F}(f))(v) = \sum_{u \in Γ(v)}f(u)$ for all $f \in F^{V(Γ)}$, $v \in V(Γ)$. In the case of finite graph $Γ$ the mapping $A^{({\rm alg})}_{Γ,F}$ is the well known operator defined by the adjacency matrix of $Γ$ (over $F$), and the theory of eigenvalues and eigenfunctions of such operator is a well-developed (at least in the case $F = \mathbb{C}$) part of the theory of finite graphs. In this paper we develope a theory of eigenvalues and eigenfunctions of $A^{({\rm alg})}_{Γ,F}$ for arbitrary infinite locally finite graphs $Γ$ (although a few results may be of interest for finite graphs) and fields $F$ with a special emphasis on the case when $Γ$ is connected with uniformly bounded vertex degrees and $F = \mathbb{C}$. By the author opinion, previous attempts in this direction were not quite satisfactory since were limited by consideration of rather special eigenfunctions and corresponding eigenvalues.
