Alon–Boppana-Type Bounds for Weighted Graphs

The unraveled ball of radius $r$ centered at a vertex $v$ in a weighted graph $G$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We present a general bound on the maximum spectral radius of unraveled balls of fixed radius in a weighted graph. The weighted degree of a vertex...

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Bibliographic Details
Published in:The Electronic journal of combinatorics 2024-02, Vol.31 (1)
Main Authors: Polyanskii, Alexander, Sadykov, Rynat
Format: Article
Language:English
Online Access:Get full text
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Summary:The unraveled ball of radius $r$ centered at a vertex $v$ in a weighted graph $G$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We present a general bound on the maximum spectral radius of unraveled balls of fixed radius in a weighted graph. The weighted degree of a vertex in a weighted graph is the sum of weights of edges incident to the vertex. A weighted graph is called regular if the weighted degrees of its vertices are the same. Using the result on unraveled balls, we prove a variation of the Alon–Boppana theorem for regular weighted graphs.
ISSN:1077-8926
1077-8926
DOI:10.37236/12212