Journal of Combinatorics

Volume 13 (2022)

Number 1

The weighted spectrum of the universal cover and an Alon–Boppana result for the normalized Laplacian

Pages: 23 – 40



Stephen J. Young (Pacific Northwest National Laboratory, Richland, Washington, U.S.A.)


We provide a lower bound for the spectral radius of the universal cover of irregular graphs in the presence of symmetric edge weights. We use this bound to derive an Alon–Boppana type bound for the second eigenvalue of the normalized Laplacian.


Alon–Boppana, normalized Laplacian, universal cover

2010 Mathematics Subject Classification

Primary 05C50. Secondary 90C35.

PNNL Information Release Number: PNNL-SA-123083.

This research was performed while the author was at University of California, San Diego.

Received 17 May 2017

Accepted 9 January 2021

Published 31 January 2022