Nonmedian Direct Products of Graphs with Loops

Abstract

A 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.