In this paper we derive a formula of the Ihara zeta function of a cone over a regular graph that involves the spectrum of the adjacency matrix of the cone. We show that the Ihara zeta function and the spectrum of the adjacency matrix of the cone determine each other and we characterize those cones that satisfy the graph theory Riemann hypothesis.

On the Ihara Zeta Function of Cones Over Regular Graphs

Zeta functions of edge-free quotients of graphs

We consider the Ihara zeta function $\zeta(u,X//G)$ and Artin-Ihara $L$-function of the quotient graph of groups $X//G$, where $G$ is a group acting on a finite graph $X$ with trivial edge stabilizers. We determine the relationship between the primes of $X$ and $X//G$ and show that $X\to X//G$ can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the $L$-function of $X//G$ evaluated at the regular representation is equal to $\zeta(u,X)$ and that $\zeta(u,X//G)$ divides $\zeta(u,X)$. We derive two-term and three-term determinant formulas for the zeta and $L$-functions, and compute several examples of $L$-functions of edge-free quotients of the tetrahedron graph $K_4$.

This announcement of results describes a ramified covering of graphs Y → X, where Y is formed by identifying a single vertex on k copies of a (q + 1)-regular graph X. We find that the reciprocal of the zeta function for X “almost divides” the reciprocal of the zeta function for Y , in the sense that the reciprocal of the zeta function of X divides the product of the reciprocal of the zeta function of Y and some polynomial of bounded degree (which depends only on the graph X). Two specific examples show that in fact “almost divisibility” is the best that can be hoped for.

Ramified Covers of Graphs and the Ihara Zeta Functions of Certain Ramified Covers.

We introduce a ramified covering of small categories, and we show three properties of the notion: the Riemann-Hurwitz formula holds for a ramified covering of finite categories, the zeta function of $C$ divides that of $\widetilde{C}$ for a ramified covering $\map{P}{\widetilde{C}}{C}$ of finite categories, and the classifying space of a $d$-fold ramified covering of small categories is also a $d$-fold ramified covering in the sense of Dold \cite{Dol86}.

Ramified coverings of small categories

“Almost divisibility” in the Ihara zeta functions of certain ramified covers of q+1-regular graphs

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Beth Malmskog

Author Tools

Chat about Author

Papers

“Almost divisibility” in the Ihara zeta functions of certain ramified covers of q+1-regular graphs

Ramified Covers of Graphs and the Ihara Zeta Functions of Certain Ramified Covers.