Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The limit theorem with respect to the matrices on non-backtracking paths of a graph

Takehiro Hasegawa, Takashi Komatsu, Norio Konno, Hayato Saigo, Seiken Saito, Iwao Sato, Shingo Sugiyama

Abstract

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.

The limit theorem with respect to the matrices on non-backtracking paths of a graph

Abstract

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.

Paper Structure

This paper contains 4 sections, 9 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The properties for the matrices on non-backtracking paths of graphs
  4. The limit theorem with respect to the matrices on non-backtracking paths of a regular graph

Key Result

Theorem 2.1

Let $G$ be a connected graph with $n$ vertices $v_1 , \ldots , v_n$ and $m$ edges. Then the reciprocal of the Ihara zeta function of $G$ is given by where $r=m-n+1$ is the first Betti number of $G$. In particular, if $G$ is a connected $(q+1)$-regular graph with $n$ vertices then

Theorems & Definitions (11)

  • Theorem 2.1: Ihara-Bass
  • Theorem 2.2: Ahumada
  • Proposition 3.1
  • Remark 3.2
  • Proposition 4.1
  • Theorem 4.2
  • Corollary 4.3
  • Theorem 4.4
  • Corollary 4.5
  • Proposition 4.6
  • ...and 1 more