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Nonmedian direct products of graphs with loops. (English) Zbl 1363.05215
Summary: 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.
##### MSC:
 05C75 Structural characterization of families of graphs 05C76 Graph operations (line graphs, products, etc.) 05C12 Distance in graphs
##### Keywords:
product of graphs; direct product; median graph