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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2208.01926 |
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| _version_ | 1866911938883092480 |
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| author | Trofimov, Vladimir I. |
| author_facet | Trofimov, Vladimir I. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_01926 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On adjacency operators of locally finite graphs Trofimov, Vladimir I. Combinatorics 05C63, 05C50 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. |
| title | On adjacency operators of locally finite graphs |
| topic | Combinatorics 05C63, 05C50 |
| url | https://arxiv.org/abs/2208.01926 |