More on the phi = beta Conjecture and Eigenvalues of Random Graph Lifts

TL;DR

This paper advances the understanding of eigenvalues in random graph lifts by exploring the conjectured equality of two categorizations of words, which could lead to improved bounds on eigenvalues related to the phi = beta conjecture.

Contribution

It proves that for every word, the categorizations phi and beta are equal when they are 2, supporting the conjecture that phi equals beta for all words.

Findings

Proves phi(w) = 2 iff beta(w) = 2 for all words w.

Supports the conjecture that phi(w) = beta(w) universally.

Provides progress towards proving Friedman's eigenvalue bound conjecture.

Abstract

Let $G$ be a connected graph, and let $\lambda_1$ and $\rho$ denote the spectral radius of $G$ and the universal cover of $G$, respectively. In \cite{Fri03}, Friedman has shown that almost every $n$-lift of $G$ has all of its new eigenvalues bounded by $O(\lambda_1^{1/2}\rho^{1/2})$. In \cite{LP10}, Linial and Puder have improved this bound to $O(\lambda_1^{1/3}\rho^{2/3})$. Friedman had conjectured that this bound can actually be improved to $\rho + o_n(1)$ (e.g., see \cite{Fri03,HLW06}). In \cite{LP10}, Linial and Puder have formulated two new categorizations of formal words, namely $\phi$ and $\beta$, which assign a non-negative integer or infinity to each word. They have shown that for every word $w$, $\phi(w) = 0$ iff $\beta(w) = 0$, and $\phi(w) = 1$ iff $\beta(w) = 1$. They have conjectured that $\phi(w) = \beta(w)$ for every word $w$, and have run extensive numerical simulations that strongly suggest that this conjecture is true. This conjecture, if proven true, gives us a very promising approach to proving a slightly weaker version of Friedman's conjecture, namely the bound $O(\rho)$ on the new eigenvalues (see \cite{LP10}). In this paper, we make further progress towards proving this important conjecture by showing that $\phi(w) = 2$ iff $\beta(w) = 2$ for every word $w$.