# Counting Graph Homomorphisms Involving Complete Graphs

## TL;DR

This paper investigates the probability of graph homomorphisms between specific graph classes, providing new bounds and insights, especially for complete graphs, and simplifies analysis by ignoring isolated vertices.

## Contribution

It introduces new theorems that establish bounds on homomorphism probabilities for complete graphs and common classes, advancing understanding in graph theory.

## Key findings

- Derived bounds on homomorphism probabilities for Kn
- Showed isolated vertices can be ignored in calculations
- Provided new theoretical results in graph homomorphism analysis

## Abstract

In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have certain desirable properties, called graph homomorphisms, and the probability of such a mapping occurring. By using notions from graph theory and combinatorics, in this paper we prove several new theorems that place bounds on this probability for certain common classes of graphs such as Kn, and show that isolated vertices may safely be ignored.