The average order of a connected vertex set in $K_m \times P_n$

TL;DR

This paper derives a closed-form formula for the average order of connected vertex sets in the Cartesian product of a complete graph and a path, providing new insights into graph connectivity properties.

Contribution

It introduces a novel closed-form expression for the average order of connected sets in $K_m imes P_n$, expanding understanding of graph product structures.

Findings

Closed-form formula for $A(K_m imes P_n)$

Enhanced understanding of connectivity in graph products

Analytical tool for graph theory applications

Abstract

Let $G$ be a connected graph. Let $N(G)$ and $S(G)$ be the number of connected sets of $G$ and the sum of the orders of these connected sets of $G$, respectively. Then $A(G)=\frac{S(G)}{N(G)}$ is called the average order of a connected set of $G$. In this paper, we derive a closed-form formula for $A(K_m \times P_n)$, where $K_m \times P_n$ is the Cartesian product of the complete graph $K_m$ and the path $P_n$.