A standard result states the direct product of two connected bipartite graphs has exactly two components. Jha, Klavžar and Zmazek proved that if one of the factors admits an automorphism that interchanges partite sets, then the components are isomorphic. They conjectured the converse to be true. We prove the converse holds if the factors are square-free. Further, we present a matrix-theoretic conjecture that, if proved, would prove the general case of the converse; if refuted, it would produce a counterexample.
Formulas for vertex eccentricity and radius for the n-fold tensor product $G = ⊗_{i=1} ⁿG_i$ of n arbitrary simple graphs $G_i$ are derived. The center of G is characterized as the union of n+1 vertex sets of form V₁×V₂×...×Vₙ, with $V_i ⊆ V(G_i)$.
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