Nonmedian Direct Products of Graphs with Loops

TL;DR

This paper characterizes median graphs formed by the direct product of graphs with loops, showing they occur only under specific path and loop configurations.

Contribution

It provides a complete classification of median graphs arising from direct products involving graphs with loops, identifying precise structural conditions.

Findings

Only path graphs with loops at an end vertex produce median graphs via direct product.

Median graphs from direct products require one factor to be a path with a loop at an end.

The characterization applies to all connected graphs with at least three vertices or with loops.

Abstract

A \emph{median graph} is a connected graph in which, for every three vertices, there exists a unique vertex $m$ lying on the geodesic between any two of the given vertices. We show that the only median graphs of the direct product $G\times H$ are formed when $G=P_k$, for any integer $k\geq 3$ and $H=P_l$, for any integer $l\geq 2$, with a loop at an end vertex, where the direct product is taken over all connected graphs $G$ on at least three vertices or at least two vertices with at least one loop, and connected graphs $H$ with at least one loop.