# Spectra of comb graphs with tails

## TL;DR

This paper conducts a detailed spectral analysis of comb graphs formed by paths and infinite rays, exploring their eigenvalues and spectral properties in finite and infinite cases.

## Contribution

It provides a comprehensive spectral analysis of comb graphs with path and ray components, including cases with infinite rays attached.

## Key findings

- Spectral properties of finite comb graphs with path components.
- Spectral analysis of comb graphs with infinite rays.
- Eigenvalue distributions in various comb graph configurations.

## Abstract

Given two graphs, a backbone and a finger, a comb product is a new graph obtained by grafting a copy of the finger into each vertex of the backbone. We study the comb graphs in the case when both components are the paths of order $n$ and $k$, respectively, as well as the above comb graphs with an infinite ray attached to some of their vertices. A detailed spectral analysis is carried out in both situations.